University of Albert a

Bezout Orders and Kru11 Rings

Enver Osmanagic O

A thesis submitted to the Faculty of Graduate Studies and Research in partial

fulfillrnent of the requirements for the degree of Doctor of Philosophy

Mat hematics.

Department of Mathematical Sciences

Edmonton, Alberta

Fa11 2000

Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliographie Services services bibliographiques

395 Wellington Street 395. me Wellington OttawaON KlAON4 Ottawa ON K I A ON4 Canada Canada

Your K k Votre reference

Our file Notre refërencn

The author has granted a non- exclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sell copies of this thesis in rnicrofonn, paper or electronic formats.

L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/film, de reproduction sur papier ou sur format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othenÿise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

Abstract

-4rithmetic rings in commutative algebra are Kru11 and Prüfer rings and therefore

include principal ideal rings, Dedekind, valuation and unique factorization rings. The

main tool in the study of arithmetic rings is an appropriate valuation theory. For

commutative domains, it is the theory of valuations on fields initiated by Krull in

1932. The most successful valuation theory for commutative rings with zero divisors is

t he theory introduced by M. E. Manis in 1967. In the non-commutativecase, Schilling

valuations tvere used to determine the Brauer group over local fields and Dubrovin

valuation rings, introduced in 1984, do not only have a rich extension theory, but are

very useful in the investigation of Bezout orders in simple artinian rings. This thesis

concentrates on the s t udy of the ideal structure of non-commutat ive vduation rings

and commutative Krull rings tvith zero divisors.

We first consider the structure of prime ideals of a Dubrovin valuation ring. -4

prime ideal P of a Dubrovin valuation ring R in a simple artinian ring Q is called

Goldie prime if R / P is a prime Goldie ring. We show that in the special case &en

R is a total valuation ring, Goldie prime ideals are exactly completely prime ideals.

Then we proceed to show that any intersection and union of Goldie prime ideals is

again Goldie prime, any idempotent proper ideal is Goldie prime and the intersection

of al1 powers of any proper ideal is Goldie prime. Using these results we show that

in the case of a rank one Dubrovin valuation ring R, the set D(R) of al1 divisorial

ideals of R is a group, order isomorphic to a subgroup of ( E t 7 +).

Next, we show that there exists a complete analogy between the structure of ideals

of a cone in a right-ordered group and the structure of ideals of a Dubrovin valuation

ring. In the r a d one czse, this structure is described completely. Also, show that

for a discrete Dubrovin voluation ring R, the Jacobson radical J ( R ) is principal as

a right R-ideal, D(R) = ( J ( R ) ) and g(R)" = (0).

Finally, we consider an application of the valuation theory in the study of com-

mutative and non-commutative arithmetic rings. If R is a commutative Krull domain

with the quotient field K, R # K, then for any h i t e set {ve,, vyn, - . - , wpn} of es-

sential valuations of the ring R and any set {ml, rn2, . . . , m,) of integers, t here exists

an element x E K such that vlri (x) = m; for all i E {1,2, . . . , n) and v V < x ) 2 O for

all other essential valuations up of R. We prove aa analogous approximation theo-

rem for KrulI rings with zero divisors. This result allows us to characterize divisorial

fractional ideals of an additively regular Krull ring.

To rny parents

To my wife Slavinka and my daughters

Tanja and Maja for their love, patience and understanding

Acknowledgement s

With deep pleasure, 1 want to express my gratitude to my supervisor Professor Ham

Brungs for his very stimulating supervision during my studies a t the University of

Alberta. His experience and broad knowledge of mathematics has aided me a great

deal in accomplishing this task. 1 am indebted to him for the time he h a . spent

introducing me to this subject, for listening and discussing my presentations, and for

his writ ten comments, suggestions and encouragement.

My sincere thanks also go to Professor Sudarshan Sehgal for his constant interest

in my progress, for his support and for the courses in Ring Theory he has taught me.

1 also thank al1 my other professors from the Department, for their effort in providing

an excellent research environment, and al1 members of my examination committee for

their consideration. 1 am also grateful to the External Examiner: Professor Günter

Krause for his comrnents and suggestions on this thesis.

My thanks go to Marion Benedict for her warm welcome, kindness, and willingness

to help.

Finally, 1 would like to express my appreciation and gratitude to my professors,

Veselin Peric and Jusuf Alajbegovic, not only for the time they had spent teaching

me algebra at the University of Sarajevo, but also for their friendship over the years.

Contents

Introduction 1

1 Preliminaries 5

1.1 EIementary properties of orders . . . . . . . . . . . . . . . . . . . . . 5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Valuation rings 13

. . . . . . . . . . . . . . . . . . . 1.3 Cones in groups and c h a h domains 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Eramples 20

2 Bezout Orders in a Simple Artinian Ring 22

') 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bezout rings -&

. . . . . . . . . . . . 2.2 The basic properties of Dubrovin valuation rings 25

. . . . . . . . . . . . . 2.3 The Ideal Tiieory of Dubrovin valuation rings 30

3 Prime Segments 51

3.1 Prime segments in Dubrovin valuation rings . . . . . . . . . . . . . . 52

3 .1 Rank one and discrete Dubrovin valuation rings . . . . . . . . . . . . 56

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Examples 61

4 Krull Rings 64

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Kru11 rings 64

. . . . . . . . . . . . . . . . . . . . 4.2 Essential valuations of Krull rings 71

4.3 An approximation theorem for Krull rings . . - . . . . . . . . . . . . 73

Bibliography 78

Index 83

Introduction

-4 successful partial proof of the famous Fermat Conjecture (before

AWiles proved the Conjecture), i-e., the impossibility of the equation xP + y p = zP

for p an odd prime number and x , y , r integers # 0, did depend on the uniqueness

of the factorization in the ring Z[Ç] ( Ç # 1 is a p t h root of unity). -4 mistake in his

first attempt to prove the Conjecture led Kummer to his study of the arithmetic of

cyclotomic fields which occupied him for almost 25 years. Using the theory of ideals

and ideal classes in algebraic number fields, Kummer proved the Fermat Conjecture

for every regular odd prime number p , i-e., p -f h, where h is the class number in O(<).

His "ideal prime numbers", that is, "the exponent" with which a factor appears in

the decomposition of a number x E Z [ j ] . is in the modern language the value of a

valvation on the field Q(C) nt x. This development of the theory of algebraic num-

bers between 1830 and 1S60 was one of the two central motivations for the creation of

modern commutative algebra and the study of commutative integral domains, prin-

cipal ideal domains ( P D ) , Prüfer, Krull, Dedekind and unique factorization domains

(UFD). The second source was Algebraic Geometry.

Even t hough a strict definition of the notion of an nrithmetic ring does not exist ,

many arguments suggest that by arithmetic rings in commutative algebra we mean

Kru11 and Prüfer rings which include principal ideal rings (PIR), Dedekind, valuation

and unique factorization rings (UFR).

The main object of study in this thesis are commutative and non-commutative

arit hnlet ic rings. Vire contribute some results concerning the ideal structure of non-

commutative valuation rings and some results about essential valuations of commu-

tative K r d rings with zero divisors.

In Chapter 1, we present background material necessary for the subject. We

discuss the basic results and present the classical theorems from the general theory

of non-commutative rings, the general theory of orders with an emphasis on Bezout

orders in an artinian ring and the theory of commutative and non-commutative valu-

ation rings. For some of these results modified proofs are given. To a certain estent,

this chapter makes the thesis independent of t h e other sources.

Ln 1984, W. Dubrovin introduced a new c l u s of non-commutative valuation rings

in a simple artinian ring, now called Dubrovin valuation rings. In Chapter 2, we

examine the ideal theory of Dubrovin valuation rings. We start with presenting

recent results by J. Grater about Bezout orders in an artinian ring and proceed to

give a modified proof of a characterization theorem for Dubrovin valuation rings

which shows that this class of rings consists exactly of local Bezout orders in simple

artinian rings. In particular, ive consider prime ideals P of a Dubrovin valuation ring

R with the property that R/P is a prime Goldie ring. Such ideals are called Goldie

prime ideals. We first show that in the special case when R is a total valuation ring,

then a prime ideal P of R is a Goldie prime if and only if P is completely prime,

see Proposition 2.3.16. Then we show that Goldie prime ideals have many %ce"

properties.

If Pi, i E A, are Goldie primes in R, then n A and U P, are Goldie prime ideals,

see Proposition 2.3.19 and Corollary 2.3.22. Also, if I2 = I # R is an idempotent

ideal of R1 then 1 is a Goldie prime, Lemma 2.3.20. The essential result for further

esploration of the ideal structure of Dubrovin valuation rings is the result in The-

orem 2-13-24. This theorem shows that the intersection of al1 powers of any proper

ideal of R is a Goldie prime.

The end of Chapter 2 looks at divisorial ideals of a Dubrovin valuation ring R. -4

right R-ideal A is called divisorial if A = -4' = n CS? where c runs over al1 elements

in Q with CS A and S = O,(A). On the set D(R) of al1 divisorial ideals ive define

the partial operation "O" by A O B = (-AB)" if A, B E D ( R ) with O,(A) = O1(B).

With respect to the operation "O", D( R) becomes an algebraic structure known as

Brandt groupoid . We show that in the case when R is a Dubrovin valuation ring of

rank one, (D( R)? O) is a group, order isomorphic to a subgroup of (R, +), the additive

group of real numbers, Theorem 2.3.15 and Lemma 2-3-25.

Chapter 3 explores the ideal structure in a Dubrovin valuation ring. In particular,

our interest lies in pairs Pl > PZ of two distinct Goldie prime ideals Pl and Pz such

that no further Goldie prime ideal exists between Pl and P2. Such a pair is called

a prime segment. We show that in a Dubrovin valuation ring there are exactly

three types of prime segments: archimedean. simple and exceptional. -4pplying this

result to a Dubrovin valuation ring R of rank one, we describe al1 possibilities for

the group D ( R ) and the subgroup H(R) of D(R) of al1 non-zero ideals which are

principal as right R-ideals. Since, in this case: for any non-zero R-ideal -4 in Q

either -4 = -4' E D(R) or A c A' and then A* = CR, A = cJ (R ) for some

c E ' ( Q ) : knowing the groups D( R) and H(R) we are able to describe completely

the structure of al1 ideals in a Dubrovin valuation ring of rank one. This result shows

that rank one Dubrovin valuation rings have an ideal structure compIetely analogous

to that of cones of right ordered groups and, equivalently, of chain domains, which

has been described earlier in papers by H.H. Brungs and G. Torner, [BT76], H.H.

Brungs and N. Dubrovin, [BD], T.V. Dubrovina and N. Dubrovin, [DD96] and H.H.

Brungs and M. Schroder, [BS95]. Also, we show that a rank one Dubrovin valuation

ring R in a simple artinian ring Q with finite dimension over its center has only an

archimedean prime segment. We conclude Chapter 3 by appiying these results to a

discrete Dubrovin valuation ring. We show that every discrete Dubrovin valuation

ring R has an archimedean prime segment J ( R ) > ( O ) , with Z ( R ) # z ( R ) 2 , the

Jacobson radical J ( R ) is principal as a right R-ideal and r) Z(R)" = (0).

The final Chapter 4 is devoted to an application of valuation theory to commuta-

tive and non-commutative arit hmetic rings. In part icular, ive consider commutative

and non-commutative Krull rings. The class of commutative ICrull rings with zero

divisors i17as introduced by R. Kenned- [IienïS] and later esplored by R. Matsuda,

[Mats-] and [MatS5]? and J. Alajbegovic and E. Osmanagic, [A090b] and [A090a].

We prove an approximation theorem for essential valuations of Iirull rings with zero

divisors: Let R be a KruU ring with total quotient ring Kt such that R # K 3 and

{vP 1 E P( R)} is the family of essent ial valuations of R. If {vY,, UV,, . . . ? vp,,}

is a finite set of essential valuations of the ring R and (ni, 7722, . . - , am,) E Zn, then

tliere esists an element r E I ï such that u T t ( z ) = mi for al1 i E {l, 2,. . . . n) and

uV(x) 2 O for al1 other essential valuations v.p of R. This results rnirrors the ana le

gous approximation theorem for essential valuations of I<rull domains. The element x

constructed in the proof of the above result is not necessarily regular. We show that

for the class of addit ively regular rings the element x can be chosen to be regular. In

this case? we give a characterization of divisorial fractional ideals in terms of principal

divisorial ideals. Also. ive hriefly discuss the notions of non-commutative Prüfer rings

and non-commutative Kru11 rings.

Chapter 1

Preliminaries

In this chapter we give the definitions and results from the general theory of as-

sociative rings which are basic for this subject. The main results are taken from

[RM87], [MMU97], [LM7l], [Sch45] and [DubS$]. Throughout this chapter, a ring is

always an associative ring with identity 1 . A subring of a ring R always contains the

identity element of R. For a ring R, we denote by U(R) the set of al1 units of R and

by Cn(0), or sometimes by Reg( R), the set of al1 regular elements of R. -4 right (left)

ideal -A of R is regular if it contains a regular element. The set of al1 zero divisors of

R is denoted by Z( R).

1.1 Elementary properties of orders

-4 ring R is called si,mple if R does not contain nonzero proper two sided ideals. If A4

is any ideal of R, R/M is a simple ring if and only if iLI is a maximal ideal. Clearly,

every division ring is simple. By Wedderburn's Theorem, if R is a simple artinian

ring then R "= Mn(D) for some division ring D.

Let C be a multiplicatively closed subset of a ring R. Then we say that R satisfies

the right Ore condition with respect to C or that C is a e h t Ore set of R if for al1

a E R and c E C, there esist 6 E R and d E C such that ad = cb. If C CR(0), then it

is called a regular right Ore set of R. .A (regular) left Ore set of R is defined similarly.

If C is a (regular) right and left Ore set of R, then it is simply called a (regular) Ore

set of R.

Theorem 1.1.1 ( [Her68]) I fC is a regular right Oir set of R, then there exists

an overring T = RC-' = & of R, called the right quotient ring of R vrith respect to

C such thnt

i) any c E C is a unit o f T ;

ii) for any q E T , there exist a E R and c E C such that q = ac-'.

For arbitraq elements ql, q 2 , . . . , qn f T there exists a common denominator, i.e., a

regular element c E C such that qi = rit-' , i = 1 , 2 , . . . , n for some ri E R.

Let R be a subring of a ring Q. If Q = RCR(0)-', then R is called a right order

in Q . Sometimes, the ring Q is denoted by Q = Q(R). Hence, a subring R of a ring

Q is a right order in Q if and only if R satisfies the right Ore condition with respect

to CR(0). .A lefi order in Q is defined similarly, and a. ring R rvhich is both a right

and a left order in Q is called an order in Q.

For a subset A of a ring R we define rR(A) = {x E R 1 -4x = O} and cal1 it

the right annihilator OF -4. The left annihilator l R ( i l ) is defined similarly. A right

annihilator of R is a right ideal of R and a left annihilator is a left ideal of R.

In an artinian ring al1 one-sided regular elements are units. More precisely the

following result holds:

Lemma 1.1.2 ( [Rob67a]) Let Q be a right artinian ring and c E Q. Then the

following conditions are equivalent:

i i) c is regular;

iii) c is a unit.

Proof. The only impiication we need to prove is (i) + (iii). Assume that rq(c) = O

and consider the chah

of right ideals of Q. Then there e x i t ~ n such that c n Q = cn+'Q. Hence: cn( l - cx) = O

for some x E Q. By (i), 1 = cx. But, cx-1 = O, implies cxc-c = O, Le., c(xc-1) = 0,

and again by (i), xc = 1, Le., c is a unit.

O

The next two results are often used in this work; they describe very useful prop-

erties of orders in an artinian ring.

Lemma 1.1.3 ( [Rob67a]) Let R be a right order in a ,right artinian ring Q

and c E R. Then, the following are equivalent:

i i) rp (c) = 0;

i i i) c 2s regular in R:

i v ) c is regular in Q;

v) c is a unit in Q .

Proof. By

let cq = O

ca = O. By

Lernma 1 .l.2, it is enough to prove (i) + (ii). Assume that rR(c) = O and

for some q E Q. Since q = ab-', where a. b E R and b is regular. then

the assumption, a = O. Hence, q = 0.

Corollary 1.1.4 ( [Rob67a]) Le t R be a right order in a right artinian ring Q

and a , b E R. T h e n ab is regular i f and on ly i f a and b are -regular.

Proof. If ab is regdar, then rR(b) = O. By Lemma 1.1.3, b is unit in Q. Then

a = (a6)b-' is regular in Q, and again by Lemma 1.1.3; a is regular in R.

17

Yote that if a ring R has a right quotient ring Q and Q satisfies ACC for right

or left annihilators, then the property in Corollary 1.1.4 holds. Rings do exits which

do not satisfy this property.

Let R be a ring and kf be a right; R-module. An R-subrnodule L of iV1 is said to

be an essential subnodule if L N # O for any nonzero R-subrnodole iV of M. If L

is an R-submodule of ibi', then there exists an R-submodule L' such that L n L' = O

and L @ L' is an essential submodule in M . If a right ideal 1 of R is an essential

R-submodule of R, then I is cailed an essential right ideal.

A right R-module U is said to be a u n i f o n n module if every nonzero R-submodule

of U is essential. A right R-module !LI is said to have finite Goldie dimension if it

contains no infinite direct sum of nonzero R submodules.

A ring R is called a right Goldie ring if R satisfies the ascending chah condition

(ACC) for right annihilators and R does not contain an infinite direct sum of nonzero

right ideals. A left Goldie ring is defined similarly, and if R is a right and left Goldie

ring, therr R is called Goldie. By Goldie's Theorem, a ring R is a (semiprime) prime

right Goldie ring if and only if R is a right order in a (semisimple) simple artinian

ring Q. Some basic properties of Goldie's rings are summarized in the following two

results:

Theorem 1.1.5 ( [RM87]) Suppose that R i s a semipr ime right Goldie ring

and let 1 be a right ideal of R -Then :

i) I is essential i f and only if I contains a regular element;

ii) i f I is essential, then I is generated b y regular elements.

Theorem 1.1.6 ( [RM87]) If a right R-module M has finite Goldie dimension,

then there ezist uniform R-submodules Ul, Uz, . . . , Un of Ad such that Ul @U2$. . .%Un

is an essential R-submodule of M . In this case n is independent of the choice of the

Ui. W e cal1 n the Goldie dimension of iW and denote it b y dR(!bl) or d ( M ) .

Definition 1.1.7 Let R be an order in a ,ring Q . Then a right R-sub~module I of

Q is called a right R-ideal of Q if

ii) there exists an element c E U ( Q ) such that CI 5 R.

A right R-ideal I of Q is called integral $ 1 C R. A left R-ideal is defined similady.

A right and lefl R-ideal of Q is called an R-ideal.

Let R be a semiprime Goldie ring and Q be the semisimple artinian ring of quo-

tients of R. If i is a right ideal of R, then 1 is a right R-ideal if and only if I is

essential. In particular, any nonzero ttvo sided ideal 1 of a prime Goldie ring R is an

R-ideal since for any nonzero right ideal X of R we have O # S I 2 X n I , i-e., I is

essential and by Theorem 1.1.5, I contains a regular element.

Let R be an order in a ring Q. For any ttvo subsets A and B of Q, tve define:

( A : B ) , = { q ~ Q 1 Bq C -4)

For a right R-ideal 1 of Q we set

Then the following holds:

Lemma 1.1.8 ( [MMUg?]) Let R 6e an order in a ring Q and I 6e a right

R-ideal of Q . Then:

i) O,([) and O r ( [ ) are orders in Q;

ii) i is a lep Or(l)-ideal and a right O,(I)-ideal;

iii) (R : I ) ! is a lep R-ideal and a right 01(1)-ideal.

Definition 1.1.9 -4 ring R is called a right (left) chuin ring i f right (leftl ideals

of R are linearly ordered b y inclusion. A right and left chain ring is called a chain

ring.

We can extend this concept in the following way:

Definition 1.1.10 Let R be a su6ring of a ring Q. Then R is called a right

n-chain ring in Q if for nny elements aa, ai , . . . , a , in Q there exists an element

ai , i E { O , 1,. . . : n ) , such that a; belongs to the right R-submodule generated 6 y the

remaining elements a i , j # i , that is, ai E a j R . A right n-chain ring in itself

is called a right n-chain ring. -4 left n-chah ring is d e h e d sirnilarly. .4 ring which

is both right and lef i n-chain m'ng is called n-chain ring.

Remark 1.1.11 a ) Let S be an o.verring of a ring R in a ring Q . If R is a right

n-chain ring in Q, then R is a right n-chain ,ring in S and S is a right n-chain

ring in Q. Furthermore, $1 C R is an ideal of Q and R is a right n-chain ring

in Q , lhen R / I is a right n-chain ring in Q / I .

b ) The class of 1-chain rings coincides with the class of chain rings.

c) In every cha in h g R, the Jacobson radical,'J(R) is a maximal right and lefi ideal.

By the Remark 1-1-11 c), if R is a semi-simple chain ring, t hen R is a division ring.

The analogous result for n-chain rings is:

Lemma 1-1-12 ( [MMU97]) Let R be a semi-simple ,ring, i.e, J ( R ) = O . Then

R is art inian if and on ly i f R is a right n-chain r ing f o r some n.

Let R be a commutative ring with identity and li its total quotient ring, and S

a multiplicative closed system of R. Then the large quotient ring of R mith respect

to S is defined to be the set RISI = {x E K 1 (3s E S) xs E R}. If -4 is an ideal of R,

then the extension of A is [AIRLq = {x E K 1 (3s E S) xs E A). A ring R is called

r-noetherian if it satisfies the ascending chah condition (ACC) for regular ideals.

Proposition 1.1.13 ( [Gri69]) For a commutat ive ring R the following condi-

t ions are eqvivalent:

ii) Every n o n e m p t y set o f regular ideals of R has a nax i rna l element;

iii) Euery regular ideal of R is finitely generated;

iv) Euety regular pr ime ideal of R is finitely generated.

-4n r-noetherian ring need not be noetherian ( [MatSl]).

Let R be a subring of a ring R' and iet a E R'. If a is a root of a monic polynomial

with coefficients from R, then a is called integral ouer R. If there exists a finitely

generated R-submodule M of R', such that an E for all n, then a is called almost

integral ouer R. The subset Ro of al1 elements in R' which are almost integral over

R is called a complete integral closure of R in R'. If & = R, we Say that R is

completely integrally closed in R'. In the case that R' is total quotient ring of R, Ro

is called a complete integral closure of R and, in the case & = R, we Say that R

is comletely integrally closed . Notions of integrally closed and integral closure are

defmed similady. If a is an integral element over R then a is aImost integral over R.

The converse does not hold. For example, T = Zp) + X Q [ [ X ] ] is a valuation domain

with value group Z x 23, and hence, T is integrally closed but T is not completely

integrally closed.

For the class of r-noet herian rings the notions of completely integrally closed and

int egrally closed coincide.

Proposition 1.1.14 ( [A090a]) [ f a r ing R is r-noetherian and integrally closed,

t h e n R is completely integrally closed.

A ring R bas few zero divisors if Z ( R ) is a finite union of prime ideals; and R is

additiueiy regular if for each r in its total quotient ring Ii, there exists u f R such

t hat r + u E CI,-(O). A ring R is called a Marot r ing or a ring wvit h the proper ty (P) if

every regular ideal is generated by reguIar elements. The relationships between these

classes of rings are summarized in the next theorem:

Theorem 1.1.15 ( [Huc88]) Consider t h e following four cond i t ions o n a corn-

,mutatz'.ve ring R

1) R is a noether ian ring;

2 ) R h a s few zero divisors;

3) R i s a n addit ively regular ring;

4 ) R i s a Maro t r ing .

None of the implications of the Theorem are reversible ( [HucSS], Examples).

1.2 Valuation rings

Valuation Theory is an important and powerful tool in many areas of mathemat-

ics: Number Theory, Theory of Local and Global Fields, Algebraic Geometry and

Model Theory. The first axiomatic definition of a valuation has been stated by Josef

KürschAk in 1913 and the further development, a t the beginning of the 30-th cen-

tury, is associated with works of Kurt Hensel, Alexander Ostrowski and Helmut Hasse.

While this stage of the development of the valuation theory is mainly concerned with

vduations of rank one7 the notion of a general valuation was introduced by Wolfgang

Kru11 in 1933, [Kru32]:

Definition 1.2.1 Let K l e a field and let V be a m a p of I< ont0 a totally ordered

abelian group C, with an added elernent m, such that CG + oa = g + ca = oo, oo > g

for al1 g E G. The map v : K -t G, U{w} is called a (Krull) valuation on Ii and

G, is called the value group of v if the foihwing conditions are satisfied

VI) (Vx E K) v(x) = oo i f and only if x = O

V3) (Vx, y E f i ) su(x + y ) 2 rnin{o(x), v ( ~ ) )

If G, = O then u is called a trivial valuation. If the group G , is the group of

integers, then u is called the discrete valuation of rank one. For exarnple, for every

x E Q = Q \ {O) and a prime number p there exist integers a, r , s such that

x = paS, where r and s are not divisible by p; then the map u, : -+ Z defined

by v,(x) = cr and u,(O) = ca is a discrete valuation of rank one of Q. The ring

& = {I E Ii [ v(x) 2 0) is called the ring of the valuation v and the prime ideal

Pu = {x E li 1 u ( x ) > O) is called the positive ideal of u. The ring R,, satisfies

the following condition: for each x E K , x E R, or x-' E R,. Conversely, if R is a

subring of K such that x E K \ R irnplies x-' E R, then there exists a valuation v of

h'such that R = &. Many authors contributed to this theory, arnong them O.F.G.

Schilling, E. Artin, O. Zariski, P.J. J d a r d , P. Ribenboim, O. Endler and M. Fuliava.

Iinill valuations axe greatly used by Zariski and his school of Algebraic Geometry.

For example: power series rings are used in the study of the algebraic varieties in

projective spaces; abstract Riemann surfaces are developed.

Among competing attempts to develop a theory of valuations for commutative

rings which may have nontrivial zero divisors, the most successful was the definition

of the valuation ring by ME. Manis in [Man67].

Definition 1.2.2 ( [Man67]) .4 map v frorn a commuta t ive ring T onto a totally

ordered abelian group G, mith co adjoined, is called a valuation of T $ the follo.wing

conditions hold:

(i) v(xy) = ~ ( x ) + v(Y);

The ring R, = {x E T 1 u(x) 2 O } is called the ring of the ualration v. The set

P, = { x E T 1 v ( x ) > O ) is a prime ideal of R, and is called the positive ideal of v.

The ring R, and the ideal P, satisfy the following condition:

(iii) For each x E T \ &, there enists y E P, such that xy E R, \ Pu.

Conversely, if R is a subring of a ring T and P is a prime ideal of R such that for each

x E T \ R there enists y E P with xy E R \ P , then there exists a Manis valuation

v on T such that R, = R and P, = P. A pair (R, P) satisfying (iii) is called a

Manis valuation pair of T . For example, the pair ( R : P ) = (Z[X], SZ[X]) is a Manis

valuation pair of the ring T = Z[X, X-'1 . If the group G, is equal to (O) , then v is

called a trivial valuation. If the group Gu is the group of integers, then v is c d e d

the diçcrete caluation of mnk one.

For the theory of Manis valuations refer to [LMiI], [HucSS] and [-4M9-1. In

the last three decades an extension of the theory of commutative integral domains

to commutative rings with zero divisors was developed, see R. Gilmer [Gil72], M.

Larsen, P. McCarthy [LMil], J. Huckaba [HucSS], R. Matsuda [MatS3] and J.

Alajbegovic, J. MoCkoI [AM92].

Io the non-commutative case, a natural question is: Wlich class of rings plays the

same role for the study of non-commutative arithmetic rings as the classes of valuation

domains and Manis valuation rings do for commutative arithmet ic rings'! First,

note that a successful "candidate" should share many properties with commutative

valuation rings, in part icular, the properties of extension of valuations. For esample,

\ire would like to know a non-commutative analogue of the following result due to

Chevalley: If V is a valuation ring of a field F and 1' is an extension of F , then

there exists a valuation ring TV of K with W n F = V. In this case we say that W is

an extension of V in K. Ln order to consider these questions one must decide what

it rneans to have a ualuation o n a division ring. It seemed natural to suggest that

this role is played by Schilling's valuations on a skew field, see (SchlS]:

Definition 1.2.3 A valuation v on a division ring D is a function from D' =

D \ { O ) onto an ordered group G,, such that for al1 a , b E D', v ( a 6 ) = v(a) + v(b)

and u(a + b) 2 min{v(a), ~ ( b ) ) .

For convenience ive extend v to D by v(0) = oo where ca > y and w + y = y+oo = oo

for al1 y E Gu. Such valuations v on D correspond to subrings R of D (called total

invariant valuation rings of D ) with the following two properties:

(T) If x E D' then x E R or x-' E R; we say that R is a total ualuation ring of D.

(1) dRd-' = R, for al1 d f O in D; we say that R is invariant .

Note that valuations on the finite dimensional centrai simple division algebras are

used to show that the Brauer groups & ( F ) over a local field F are isomorphic to

Q/Z. Using valuations. the central simple algebras over global fields: i-e., algebraic

number fields or algebraic function fields, are classified. This work is associated with

the famous names of H. Hasse, R. Brauer, E. Noether and A. Albert, see [PieS-].

However, the valuation of the center F of a division ring D such that [D : FI < oo, in

general, cannot be extended to the whole algebra. For example, for p an odd prime

number, the valuation ring Z(,) in the center Q = Z(W(Q)) of the Cdimensional

algebra of quaternions over the rationals cannot be extended by a total invariant

valuation ring in E(Q), see Example 1.42. This difficulty has been overcorne by the

introduction of Dubrovin valuation rings in a simple artinian ring, see [DubS1]. A

very well developed ideal theory and extension theory in the finite dimensional case

justify the name valuation rings.

Definition 1.2.4 Let R be a svbring in a simple artinian ring Q and asswme that

M is an idenl of R such thn t R / M is a simple artinian ring, and /'or each q E Q \ R

there are ele,ments r ,rr E R with qr,rrq E R \ M . Then R is called a Dubrovin

valuation ,ring of Q.

Exarnples of Dubrovin valuation rings are given in Section 1.4.

Note: that in the case Q = K is a commutative field the classes of Schilling val-

uation rings. i.e, total invariant rings in Q, total valuation rings in Q and Dubrovin

valuation rings of Q al1 agree and are in fact equal to the class of the classical corn-

mutative valuation domains in Q = K. More precisely:

Theorem 1.2.5 Let R be a Berout order in a simple artinian ring Q. Then:

1. R is a Dubrovin valuation ring o f Q i f and only if R / J ( R ) is a simple artinian

ring.

2. R is a total valuation r ing of Q i f and only i f R / J ( R ) and Q are skew fields.

3. R is a Krull valuation r ing in Q i f and only i f R / Z ( R ) and Q are fields.

For the general theory of Dubrovin valuation rings see [DubM], [DubSS], [Dubgla],

[GraSzb] and [MMU9'7].

We mention here the foilowing extension t heorem for Dubrovin valuation rings.

Theorem 1.2.6 ( [Dub85] and [BG90]) F o r euery valuation ring V in the

center F of a jînite dZ.mensional central simple algebra D, there exists a Dubrouin

valuation ring R of D such that R n F = V .

A very rich extension theory using Dubrovin valuation rings has been developed

in the last decade, see for example papers by PI'. Dubrovin [DubSj], H.H. Brungs, J.

Grater [BGSO], J. Grater [Gra92a], P. Morandi [MorSS], P. Morandi, A. Wadsworth

[MW891 and A. Wadsn-orth [WadSS]. This suggests that among three competing

concepts of noncommutative valuation rings, Le., Schilling's total invariant valuation

rings, total valuation rings in a skeew field (equivalently, chain domains) and Dubrovin

valuation rings in a simple art inian ring, the last one is the most effective for the study

1.3 Cones in groups and chain domains

Let G be a group. A cone of a group G is a subset P of G such that PP C P and

P U P-' = G. It follows that the identity element e of G belongs to P. If in addition

P n P-' = {el, then P is called a basic cone of G. In this case, we Say that (G, P)

is a right ordered group with a 5, b if and only if ba-l E P for a, b E G. The right

order <, will also be a left order if and only if aPadl = P for all a E G. The group

(G, P) is then a linearly ordered group.

Let P be a cone of G. -4 non-empty subset 1 of G is called a right P-ideal if

I P C 1 and 1 Ç aP for some a E G. Left P-ideals and P-ideals are defined similarly

If 1 Ç P then we omit the prefix "P-" and cal1 I a right (left) ideal of P. Moreover,

if 1 # P, then we Say that (right or left) ideal 1 is proper. A proper ideal 1 of the

cone P is said to be pr ime (completely p r i m e ) if for a , b E P we have a E 1 or 6 f 1

whenever a Pb C I (respectively, ab E I).

The proofs of the following resuIts are similar to those of the corresponding as-

sertions about chain rings:

C l ) The set of right (left) P-ideals is totally ordered with respect to inclusion.

C2) The set of elements invertible in the cone P is the intersection P n P-' and it

is a subgroup of G.

C 3 ) The subset J ( P ) of elements non-invertible in P is a maximal right (left)

ideal. This ideal is a completely prime ideal and P = J ( P ) U U ( P ) and

U ( P ) r) J ( P ) = 0, where U ( P ) is the set of a11 units of P.

We note that a s ~ b n n g R of a skew field D is a total ualuation ring $ and only if (Ra, - ) is a cone in ( D a , -). -4ny total valuation s u b n n g R of a skew field D is a

Bezout order in D with R / J ( R ) a skew field.

Let G be a group with a cone P. We Say that a total valuation ring R of a skem

field D is associated with P if the following conditions hold:

b) Every element d E D' can be written as d = glu1 = u2g2 with gl, g~ E G and

ul, 112 E LJ(R) and PgI P = Pg2 P ;

The existence of total valuation rings wit h prescribed associated cones has been

studied recently, see papers by H.H. Brungs and G. Torner, [BTSS], T.V. Dubrovina

and N.I. Dubrovin, [DD96], and H.H. Brungs and N.I. Dubrovin, [BD]. Many

examples are constructed by the authors mentioned above.

The following result describes the correspondence between the c h i n of right P-

ideafs in a group G with a cone P and the chain of right R-ideals in the total valuation

ring R of a skew field D associated with a cone P.

Theorem 1.3.1 Let a total valuation ring R o f a skew field D be associated with a

cone P of a group G. T h e n ~ ( 1 ) = IR, for a right P-ideal 1, defines a n iso~morphisrn

between the chain o f right P-ideals in G and the chain of right R-ideals in D. T h e

inverse mapping .ri, assigns t o each right R-ideal A the right P- ideal +(.il) = A n G.

This correspondence preserues the properties of being a n ideal, a completely prime

ideal and a prime ideal, and R-ideals correspond to P-ideals.

Since the cone R' of D' is associated with the total valuation ring R in the skew

field D, an immediate consequence of Theorem 1.3.1 is the following result:

Corollary 1.3.2 Let R be a total ualuntion subring of a skew field D wiih right

(lefi) R-ideal A. T h e n A \ { O } is a right (lefi) R'-ideal i n D'. Conuersely, if 1

is a right ( l e m R'-ideai i n D*! then I U { O ) is a right ( l e p ) R-ideal i n LI. T h i s

correspondence preserues the properties of being an ideal, a pr ime ideal, a co,mpletely

prime ideal and o f being proper.

We conclude this section with the result by N-Dubrovin, which will allow the

construction of total valuation rings wit h prescribed associated cones.

Theorem 1.3.3 ( [Dub93]) Let P Se a cone i n a group G' and F a skeu; field

such that the following condit ions hold:

a) P does n o t con ta in a m i n i m a l ideal;

b) For every right P-ideal 1, the group ring F U ( O r ( I ) ) is an Ore domain.

Then there exists a skew field D and a total ualuation ,ring S of D associated .with P.

1.4 Examples

Wë have seen in Theorem 1-2.5 that if R is a Dubrovin valuation ring in a simple

artinian ring Q with R / J ( R ) a skew field, then Q is a skew field and R is a chain

domain. Hence, chain domains are special Dubrovin valuation rings. Also, rnatrix

rings over chain domains are Dubrovin valuation rings.

A particular esample of a Dubrovin valuation ring is the following:

Example 1.4.1

The r ing R = hl(Z(,)) of quaternions over the valuation ring Z(,), cuhere p is an

odd prime number, is a Dubrovin valuation ring in the ring Q = H(Q) of quaternions

ouer the rationals.

Proof. M = pR is a two sided ideal of R, R / M = Q @ FPi @ IFp j @ Fpk, where Fp

is a field of p elements. Since R/M contains nontrivial zero divisors, we have R / M 2

MZ(Fp). TO check the second condition of Definition 1.2.4 we take an arbitrary

element q = q~ + qli + qz j + q3k E Q \ R. Shen at least one of qi does not belong

to Z(,). Among these, we choose one, Say q,, ivith smallest pad ic valuation. Then

qs = A p"st, where a, > O and p f r,, p { t,. Let r = p a S t , E R. Then qr E R \ M.

The nest esample was announced in Section 1.2.

Example 1.4.2 A ualuation domain V of the center F of a central simple diuision

algebra D with [D : F ] < co, cannot be extended to a total ualuation ring in D.

Proof. Let p be an odd prime number, D = H(Q) be the Pdimensional algebra of

quaternions over the rationals, F = Q is the center of D and V = Z(,) the valuation

ring of the padic valuation on F = Q. Assume thât t k e exkts a total vduation ring

R in D which extends V' i-e., R 0 Q = V. Since R is a chain domain, the Jacobson

radical ikf = J ( R ) is a maximal right (left) ideal of R. Hence, R/M is a division ring.

Also [R/M : Z~,)/pZ(,J 5 [D : Q] = 1 < 00. Therefore, R/M is a finite division ring .. .

and b - Wedderburn Theorem, R/M is commutative. So = ,i.e., 2ji E M. But:

i,j are units in D. Hence, 2 E M , a contradiction which shows that V = Z(,) cannot

be extended to a total valuation ring in D.

O

Exarnple 1.4.3

The valuation Rng V = Zp) of Q has a total invan'ant extension in D = H(Q).

Proof. For an element a = a0 + a l i + a z j + a& E D , where a; E 0, we define

a' = a0 - al i - a 2 j - a3k and N ( a ) = aa* = a: + a: + a$ +a$. It can be shown that:

1. ( a + a)= = al + O-. (a0)- = 9%-, for al1 a , j3 E 0;

2. N(c@) = N(a)N(,O), for ail a: ,O E D;

3. N(a - l ) = N ( a ) - ' , for al1 a E D;

4. B = {a E D 1 N ( a ) E Z(2)) is a subring of LI;

Now. for a E D \ B, N ( a ) E Q\ Z(2). Hence, N ( a - l ) = N(a)-' E Z(2). So, a-' E B,

Le., B is a total valuation ring in D. Also, crB = Ba for al1 a E D, Le., B is invariant.

Hence, B is a total invariant extension of V in D.

El

Chapter 2

Bezout Orders in a Simple

Artinian Ring

A ring R is said to be local if R = R / J ( R ) is a simple artinian ring. In this chapter

we consider Bezout rings, and, in particular, local Bezout orders in a simple artinian

ring. These orders are exactly Dubrovin valuation rings (see Theorem 2 2 s ) . In

Section 2.3 ., the prime spect rum of a Dubrovin valuation ring is investigated.

2.1 Bezout rings

A commutative integral domain is called a Bezout domain if each finitely generated

ideal of R is principal. Hence, every principal ideal domain (P ID) is a Bezout domain

and every noetherian Bezout domain is a PID.

ExampIe 2.1.1 There ezists a Bezout do.main which is not a PID.

Proof. Let (G, P ) be a commutative ordered group with the positive cone P, i.e.,

P P Ç P, P U P-' = C: and P n P-' = { e ) . Let F be any field and consider

the group ring FG and the subring T = FP = { C a , p 1 a, E F, p E P}. Then

S = {C a,p 1 a. # O) c T is a multiplicatively closed system in T. Since F is a field

and G' is a commutative ordered group, FG does not have nontrivial zero divisors

and it is a commutative dornain. Consider the localization V = TS-' of T a t S in

the quotient field K = Q ( F G ) of the domain FG.

The set of al1 nonzero principal ideals of V is exactly the set { g V 1 g E P ) and

glV = g2V if and only if gl = 92.

For, let O # t = a,g~ + a,,gi + - + a,,g, E T = FP where gi E P, a , E F\ { O h be any element and assume that e < go < 91 < 9 2 < - . < gn. Then t = gos1, where

S I = ~,+a, ,g; '~~+- -+a,,g~'g, E S. Since V = TS-', each element O # v E V has

the form v = ts-' for some t E T and s E S-Hence, vV = gosls-'V = gQV.Assume

g lV = g2V but gl < 9 2 . Then V = g ; 1 g 2 ~ = p V for some p E P.This implies

1 = pts-', i.e.: s = pt E S, a contradiction.

Hence, the lattice of principal ideals of V is a chain, and if I = (vl : us,. - .un)V

is a finitely generated ideal of V, then I is principal, Le., V is a Bezout domain.

Note that V is a valuation domain and the valuation associated to V on I< has G as

the value group. Choosing the group G to be (IR, +), the valuation domain V is not

discrete and hence, V' is not a PID.

O

Every Bezout domain is a Prüfer domain, i.e., every finitely generated ideal of R

is invertible. Therefore, for each commutative Bezout dornain R and a prime ideal

P of R, the localization Rp is a valuation ring and R = n Rp, where P runs over al1

maximal ideals of R.

Noncommutative Bezout rings have been studied by many authors, see P. Cohn

[Coh63], [CohS5], J. Robson [Robgïb], H.H. Brungs [BruS6], R. Beauregard [Bea'iS].

Following [RobG'ib], a ring R is called a right Bezout ring if the following conditions

are satisfied:

i) a b is regular in R if and only if a and b are regular;

ii) a R n bR contains a regular element for every pair of regular elements a, b of R;

iii) For every pair of regular elements a, 6 E R the right ideals a R n bR and aR + b R

are principal.

A left Bezout ring is defined similarly and a Bezout ring is a ring which is right and

left Bezout. A right Bezout ring is not necessarily a left Bezout ring but if a ring R

has a left and right quotient ring Q which is ar t idan, then R is right Bezout if and

only if R is left Bezout. Also, $ R is sernipnrne Goldie, then R is a Bemut ring if and only if e v e q finitely generated right ideal of R is principal, see [RobG'ib], Th.2.4.

and Th.3.5.

Example 2.1.2 There exists a ring R that is a right Bezout ring but it is not a

left Bezout ring.

Proof. Consider the field Ii = Q(t) and define the map a : K +- K with ~ ( q ) = q,

for al1 q E Q and g ( t ) = t2. Then o is a monornorphisrn from K into Ii. Define

R := {CE0 xiai 1 ai E ri) where ax = zo(a) , for al1 a E K. Then R is a ring.

An element O # r = a0 + tal + x2az + - - E R is a unit if and only if a0 # O. Let

f = xna, + ~"+'a,+~ + . . E R where n 2 1 and a, # O. Then f R = xnuR, where

u = a, + xa,+l + - E U ( R ) , i-e., fR = xnR. Hence, al1 principal right ideals of R

are of the form xnR, n = 1, 2, - . . , Le., R is a right Bezout ring.

On t.he otlier hand, for the left ideals Rx and R x t , the intersection Rx n Rxt = 0 ,

i.e., R is not a left Bezout ring.

0

In [Bea73] it is shown that any ring T between a Bezout domain R and its

quotient field Ii is a quotient ring of R and T is also a Bezout domain. The following

theorem generalizes this result .

Theorem 2.1.3 ( [Gra92b]) Let R be a fiight) Berout ring having a right and

left quotient ring Q which is right artinian and let T be an ovem-ng of R in Q.

Furthemore, let S = U ( T ) n R. Then the following hold:

i) S is a left Ore set of R;

i i ) T

iii) T is a Bezout ring

2.2 The basic properties of Dubrovin valuation rings

Throughout this section, Q is a simple artinian ring, R is a Dubrovin valuation ring of

Q and k.1 is the ideal satisfying conditions of Definition 1.2.4. The following results

are the main steps to pro= that !LI is the Jacobson radical of R and that it is a

unique maximal ideal of R. The results are important by themselves and very often

used in this work.

Lemma 2.2.1 ( [Dub84]) For any right (lefi) R-subrnodule I of Q the following

holds:

Proof. Let I be a right R-submodule of Q so that I n R C M. Let x E 1. If

x E Q \ R, then there exists an element r E R such that xr E R \ Ad. But, xr f Il

i.e.,zr E I r ) R C M , acontradiction. So, I E R, i.e., 1 Ç R. Hence, I = I n R 2 M .

Lemma 2.2.2 ( [Dub84]) M does not contain nonrero right (lep) ideals of the

ring Q.

Proof. Let a E Q, a # O and aQ M . I f I + aQ C U ( Q ) , then aQ C J ( Q ) = O ,

Le., a = O. So: 1 + aQ U ( Q ) . Therefore, there exists an element q E Q such that

1 + aq 6 CI(Q). But, Q is an artinian ring and by Lemma 1.1.2, every one sided

regular element of Q is invertible. So, L := Iq( 1 + aq) # O. Also, L 0 R C M. Hence,

by Lernma 2.2.1, L Ç M. Since Laq # O, LaQ is a nonzero 2-sided ideal of the ring

Q. Hence, LaQ = Q. Then 1 E LaQ C bf, a contradiction.

cl

Lemrna 2.2.3 ( [Dub84]) 1 + M 2 u ( R )

Proof. Let m E LW and L := Zq(l + m). Then L n R E M. By Lemma 2.2.1 and

Lemma 2.2.2, L = O. Since Q is an artinian ring, 1 + rn E U ( Q ) . So, for any rn E M ,

( 1 + m)-' exists. If ( 1 + m)-' E Q \ R, then (1 + m)-' - 1 6 R and there exists an

element s E R such that r := [(l + m)-' - 11s E R \ M. Then ( 1 + m)r = -ms, so

that r = -m(r + s ) E M , a contradiction.

Theorem 2.2.4 ( [Dub84]) Let R be a Dubrouin valuation ring of Q . Then LW

is the Jacobson radical of R. In particular: hl is the unique maximal ideal of R.

Proof. Since R / M is a simple artinian ring , f(R) E M. By Lemma 2.2.3, 1M

Corollary 2.2.5 Let S be an ouerrzhg of R in Q and le t 1 be a proper ideal of

S. Then 1 C Z ( R ) . In particdar, J ( S ) Ç y ( R )

Proof. Let 1 # S be any two-sided ideal of S. Then I n R # R, i x . , I r ) R is a

proper ideal of R. Hence, by Theorem 2.2.4, I n R c g ( R ) and by Lemma 2.2.1;

1 c J ( R ) .

To characterize Dubrovin valuation rings in terms of Bezout orders and n-chah

rings, we need the next two results.

Lemma 2.2.6 ( [Dub84]) Let R be a Dubrovin uahation ring of Q and I be a

right R-submodule of Q. Then

Proof. First note that I / I J ( R ) is a right R-module. Let n = dR(R) be the Goldie di-

mension of the R-module R a n d assume that the opposite holds, Le., d E ( I / I f ( R ) ) >

n. Then, since R is a simple artinian ring there exist a , b E 1, b I g ( R ) such that

and

since ( a R + I J ( R ) ) / I g ( R ) 9 a R / a R n I Y ( R ) and a J ( R ) C a R n I g ( R ) , it follows

from aR/aR n I T ( R ) R / g ( R ) that a R n I g ( R ) = a r ( R ) and hence, aR/aJ (R) E

R / Y ( R ) - So, r ~ ( a ) C J ( R ) . On the other hand, rR(a) = rQ(a) n R. Hence, by

Lemma 2.2.1, rQ(a) C Z ( R ) . Then, by Lemma 2 - 2 2 and Lemma 1.1.2: a is a unit

in Q. So, we can rewrite aR n I J ( R ) = a z ( R ) in the form R n a- ' IJ(R) = J ( R ) .

Again, by Lemma 2.2.1? a- ' IJ (R) Ç z(R), i-e., I J ( R ) = a J ( R ) . Now,

Hence. a R n bR C a J ( R ) , i.e.? R n a-'bR g ( R ) . By Lemma 2.2.1: a-'bR C J ( R ) .

This implies b E I g ( R) , a contradiction.

O

Lemma 2.2.7 ( [Dub84]) Let R be a Dubrovin valuation ring of Q. Then any

finitely generated right (left) R-subrnodule of Q is principal.

Proof. Let I be a right finitely generated R-submodule of Q. By Lemma 2.2.6,

dR(l/IJ(R)) 5 dR(R), where R = R / Z ( R ) is a simple artinian ring. Hence, R =

I / IJ(R) @ N for some right R-submodule N of R. Since R is a cyclic R-module,

I/IJ(R) is a cyclic R-module and hence 1 / 1 3 ( R ) is a cyclic R-module, i-e., aR + IJ(R) = I , for some a E 1. But, 1 is a finitely generated R-module. By Nakayama7s

lemma, this implies that I = aR, i.e., 1 is a cyclic R-module. SimiIarly for left

R-submodules .

O

The next result is the main characterization theorem for Dubrovin valuation rings.

Theorem 2.2.8 ( [DubS4] and [MMU97]) Let R be a subring of a simple

artinian ring Q. Then the following conditions are equiualent:

(1) R is a Dubrovin ualuation ring O/ Q.

(2) R is a local Bezout order in Q.

(3) R is a local n-chain ring in Q for some n with d(R) 3 n, vrhere R = R/J(R).

Proof.

If Ive assume that (1) holds, then by Lemma 2.2.7, R is a local Bezout ring.

Let q E Q be any element. Consider the finitely generated right R-submodule

R + qR of Q. Then R + qR is principal, i.e., R + qR = SR for some s E Q. Hence,

1 = sr for r E R. This implies that the right annihilator rp(r) of r in Q is equal to

0, rq( r ) = O. Hence, by Lemma 1.1.2, r is a unit in Q. On the other hand, q = sx

for some x E R. So, q = r-lx.

We are left to prove that every regular element in R is invertible in Q. Let c E R

be a regular element and consider q E ro(c). Then, cq = O. If q E Q \ R, then by (l),

there exists an element r E R so that qr E R \ 3 ( R ) . Then cqr = O and qr E R irnply

that qr = O, since c is regular in R. Hence, qr f J ( R ) , a contradiction which proves

that q E R, and hence, q = O. So, r Q ( c ) = O, and by Lemma 1.1.2, c is a unit in Q.

This proves that R is a left order in Q. Similarly, R is a right order in Q.

Assume that (2) holds and let n = d(@. Let qo, q,, - - , qn E Q. Then goR + Q I R + - - - + qn R is a finitely generated right R-submodule of Q. By (2 ) : we can write

-1 qi = si ri7 where ri, si E R with si regular elements. For elernents s ~ ' E Q, there

-1- exist a common denominator, Le., a regular element t E R such that s ~ ' = t r,,

f o r s o m e r ; ~ R. T h e n q o R + q 1 R + - - + q , R = t - L ( i ; o r o R + ~ r l R + . - - + ~ r n R ) .

Since R is a Bezout order in a simple artinian ring Q, the finitely generated right

ideal %roll + Krl R + . . + Kr, R of R is principal. So, there exists an element s E Q

so that qoR + ql R + - + q,R = SR. Since d ( s R / s g ( R ) ) 5 n = d ( R ) , there exists

i, Say i = O, such that qlR + --• + qnR + s J ( R ) = SR. By Nakayama's lemma,

ql R + - - - + qnR = SR. So, qo E SR = q, R + . - . + qn R, Le., R is a right n-chah ring

in Q. Similarly, R is a left n-chah ring in Q.

Suppose that (3) holds and let rn = d ( R ) , i.e., rn is the length of "the longest" direct

sum of uniform right (left) ideds of R which is essential. We only need to prove that

for al1 q c Q \ R there exist r , r' E R such that qr, r'q E R \ z ( R ) .

Let q E Q \ R. Since R is a simple artinian ring, right ideals of R are generated - by idempotents. Hence, there exist a set {q7 G, - - - , e,) of primitive orthogonal

- idempotents of R. Let el, e2, , e, be the inverse images of c, si - - , e, under

the map R t R = R / J ( R ) respectively. By Remark 1-1-11 a), R is an n-chain - ring. Since m 2 n, R is an rn-chain ring. Consider the elements q, c7 G, . . - , e,.

There are two possibilities, qR + J ( R ) E CE, e;R + g(R) or there exists i such that

eiR C qR + x e j R + J ( R ) . The first case is impossible, since then q E R. Hence, i f i

e; = qr+ Cejr j+x, where x E z ( R ) and rj , r E R. Then qr E R\Z(R). Otherwise, jf i -

ei - Cejrj E y( R): i-e., = Cej+io7 i.e., the idempotents {q, c, - - - , e,) are not i# j j#i

primitive.

Similarly, there exists r' E R such that r'q f R \ g(R).

Note that every condition in Theorem '2.2.5 is equivalent to the condition that R

is a local semi-hereditary order in Q. Since the notion of semi-hereditary order is not

essential for this work we omit the details of this characterization- For details see

2.3 The Ideal Theory of Dubrovin valuation rings

Throughout this section, R is a Dubrovin valuation ring i n a simple artinian ring Q.

In this section we study the properties of R-ideals, divisorial ideals and overrings of

R. In particular, ive consider prime ideals of R. The foliowing result is crucial for

the study of the ideal theory of Dubrovin valuation rings- The detailed proof can be

found in [DubS$] and [MMU97].

Lemma 2.3.1 ( [Dub84] and [MMU97]) Let R be Du6rovin valuation ring

of Q and let T2 C Tl be right R-subrnodules of Q such that

(1) Tl ZS regular and

( 2 ) there ezists a sulring S of 01(T2) such that for an y regular elements t l , t z E Tl

there is a regular element t E Tl with Stl + St2 C - St.

Then either Tl = T2 or there is a regular element to E Tl such that Tz C t o J (R ) .

The next result shows that Dubrovin valuation rings share a very important prop-

erty with other types of valuation rings.

Proposition 2.3.2 ( [Dub84]) Let R be a Dubrouin valuetion ring of Q and

let S' be a Beaout order in Q. Then the set of regular S - R-sub-bimodules of Q is

linearly ordered by inclusion. In particular, the set of al1 R-ideals of Q and hence,

the set of ail two-sided ideals of RI are linearly ordered b y inclusion.

Proof. Let Tl , T2 be regular S - R-sub-bimodules of Q. Set To = Tl n 37% If To = Tl ,

then Tl Ç T;?. Let TO C Tl. Then, by Lemma 2.3.1 there exists a regular element to E

Ti such that TO C t d ( R ) c toR C T I . Hence, Tz r) toR C T2 n Tl = To 2 t o 3 ( R ) ,

that is? 1;'- n R Ç J ( R ) . B y Lemma 2.2.1, tr1T2 C J ( R ) , Le- Tî C t O J ( R ) C Tl-

17

Let P be a prime ideal of a ring R. If CR(P) := { r E R ( r + P is regular in R / P )

is a regular Ore set of R, then the quotient ring RCR(P)-' = CR(P)-l R of R with

respect to CR(P) is denoted by Rp =PR and is called the localization of R. The

overrings T of a Dubrovin valuation ring R in Q are again Dubrovin valuation rings

of Q that are in one-to-one correspondence with the prime ideals P of R for which

R / P is prime Goldie. More generally, the following results hold:

Theorem 2.3.3 ( [Dub84], [Gr592b]) Let R 6e a Bezout order in a simple

artinian ring QI let B be a Dubrouin uahation ring of Q contaitzing R and let P =

J ( B ) R. Then P is a prime ideal of R such that R/P is Goldie, CR(P) = {r E

R 1 r + P is regular in R /P ) is a regular left and right Ore set and B = Rp =PR.

Conversely, $ R is a Bezout order in a s imp le ar t inian r ing Q and P is a prime

ideal o f R such tha t RI P is Goldie, t hen CR(P) is a regular lef i and n g h t Ore set and

Rp is a Dubrouin ualuation ring o f Q s u c h that J ( R p ) n R = P.

Theorem 2.3.4 ( [Dub84]) Let R be a Dubrouin ualuation ring of a simple

ar t inian ring Q and let S be a n ouerring of R in Q . T h e n R = R / J ( S ) is a Dubrovin

v a h a t i o n ring of 3 = S / g ( S ) , S i s a Dubrovin ualuation r ing of Q and S = Ry(s).

Proof. By Corollary 2.2.5, J ( S ) C R and by Theorem 2.2.S, R is an n-chain ring

in Q with d ( R ) = d ( R / J ( R ) ) 2 n. Hence, by Remark 1-1-11, S is an n-chain ring

in Q and S is an n-chain ring in S. Since S is a semi-simple ring, by Lemma 1.1.12, - S is an artinian ring. If I C S is any ideal of S, then by Corollary 2.2.5; 1 C r ( R )

and hence, 1 + I C 1 + 3 ( R ) 2 U(R) C U ( S ) , i.e., 1 Z ( S ) . So, = S / Z ( S ) is

a simple artinian ring . By Theorem 2.1.3, S is a Bezout order in Q. Hence, S is

a Dubrovin valuation ring of Q. Since R / r ( E ) P R / J ( R ) is a simple artinian ring

and d ( Z / ~ ( ? i ) ) > n, by Theorern 2.2.8, R is a Dubrovin vduâtion ring in S. Hence,

R / J ( S ) is a prime Goldie ring and by Theorem 2.3.3, S = Rs(s).

0

It follows, that prime ideals P of a Dubrovin valuation ring R such that RI P is a

prime Goldie ring, are very important. Therefore, the following definition is natural:

Definition 2.3.5 T h e pr ime ideal P of a Dubrovin ualuation r ing R is called a

Goldie p n m e ideal if RI P is a prime Goldie ring.

Xote that J ( R ) and (0) are Goldie primes of a Dubrovin valuation ring R.

Remark 2.3.6 I n the special cases, when R is a total valuation ring in a skew

field Q, i.e.! R is a Bezout order in Q with R / J ( R ) being a skew field. o r when R is

a Bezout order in a central sinmple algebra Q , the bcect ion given in Theorem 2.3.3,

is described by the following results:

a) If R is a Bezout order i n a skew field Q such that R / J ( R ) is a skew field, Le.,

R is a total valuation n'ng in Q , then there exists a one-to-one correspondence

between overrings B of R i n Q and the set of completely prime ideals of R given

b y B + J ( B ) n R a n d P + R S d L = Rp, w h e r e S = R\P.

b) ( [Gra92b]) If R i s a Bezout order in a central simple algebra Q , B is the set of

all Dubrouin vahat ion rings of Q containing R, and P is the set of al1 prime

ideals of R , then the n a p f : B + P given by B ct J ( B ) n R is a uel l defined

anti-order isomorphism iuhere f - l ( P ) = Rp- hfooreouer, R = n Rp where the

intersection .mns over al1 maximal ideals of R, i.e., R is a n intersection of

Dubrocin valuation rings.

Remark 2.3.7 From the result in Remark 2 3 . 6 b), it follows that if R is a

Dubro.uin valuation n'ng in a simple art inian ring Q with finite dimension over i ts

center Ii', then euery prime ideal P of R is Goldie prime and there is a bijection

between specR and the set of al1 ouerrings of R i n Q given by P Rp ( R p i s the

localization with respect to the set CR(P) of elements r E R regular modulo P l . Th i s

result was obtained b y Dubrouin, [DubS5], Theorem 1.

In the following part of this section we discus a theory of divisors of a Dubrovin

valuation ring. The following results are needed.

Lemma 2.3.8 ( [MMU97]) Let R 6e a Dubrovin valuation ring of Q. Then

O , ( J ( R ) ) = Or(Z (R) ) = R.

Proof. Assume R C O,(J( R)). Then by Lemma 2.3.1, there exists a regular element

to E O,(g(R) ) such that R 2 t J ( R ) and hence, ta1 E J ( R ) . But then, 1 E J ( R ) ,

a contradiction. Thus, R = O,(Z(R)). Similarly, R = Oi(Z(R)).

Lemma 2.3.9 Let R be a Dubrovin ualuation r ing o f Q a n d P be a p r i m e ideal of

R. Then O,(P) = Or(P).

Proof. Since the set of al1 overrings of R in Q is linearly ordered by inclusion, we may

assume Of( P) C Or(P)-Then, JyOr(P)) C 3(Ol(P)) . Hence, J (Or (P) ) (Or ( P ) P ) =

(J (O,(P))O,(P))P C 3 ( O r ( P ) ) P E J ( O i ( P ) ) P C O I ( P ) P C P. But 3 ( O r ( P ) )

and O,(P)P are 2-sided ideals of R. Thus, since P is a prime ideal of R, we have

Z(O,(P)) E P or O,(P)P C P, Le, P = J (O , (P ) ) or O,(P) = Or(P) . Thus,

Oi(P) = Oi(z(O,(P)) = O,(J(O,(P)) = Or(P) by Lemma 2.3.S, or O,(P) = Ol(P).

So, in both cases Or( P) = Or(P).

Lemma 2.3.10 ( [MMU97]) Let R be a Dsbrovin ualuation ring of Q , -4 be a n

R-ideal of Q and le t S = O,(A) . T h e n the following are equiualent:

(1) -4 is principal as a right S-ideal.

Proof.

(1) (2 ) : Let A = aS for some a E A. Then A-' = Sa-' and (2) follows.

( 2 ) (3): Let A-'-4 = S. If A = A z ( S ) , then -4-'A = A-'A J(S), i.e., S = J(S),

a contradiction.

(3 ) + (1): Let A > rlZ(S). Then by Lemma 2.3.1, there exists a regular element

t E -4 such that -4,7(S) C tZ(S) C A z ( S ) , that is, tJ(S) = AJ(S). Hence,

t-'AJ(S) = J ( S ) . Therefore, t-'A C 0 i ( 3 ( S ) ) = S. Thus, A E tS , that is,

A = ts. O

On the set of right (left) R-ideals we introduce the following operation:

Definition 2.3.11 Let 1 be a right R-ideal and S = O,(I). LVe de jne 1' = n CS, where c runs over all elements in Q with CS > i. Similady, for any left R-ideal L

with T = O I ( L ) we define *L = nTc , where c r u n s over all elernents in Q with

T c L.

That 1' =*I for an R-ideal I follows from Proposition 3-3-13. In the next result,

the basic properties of the "*"- operation are given. Note that in Chapter 4, we

briefly discuss an analogous operation for commutative rings.

Proposition 2.3.12 ( [MMU97]) Let R be a Dubrouin valuation ring of Q and

let 1 be a e h t R-ideal o f Q . Then:

(3) (ci)' = ci', for any c E U(Q)

(4) (CI)-' = I-'c-~, jOr any c E U ( Q )

Proof. (1) : This is trivial.

(2) : By (l), I' C (1')'. Assume 1' c (1')'. Then, there exists an element

x E (IR)' such that x $- 1'. Therefore, there exists an element bo E Q with boS > 1 but x $ boS. Also, x E CS, for al1 c E Q with CS 2 IV. But, boS > 1 implies 1- Ç boS

and thus, x E b05, a contradiction.

(3) : Let c E U ( Q ) and x E 1' be an arbitrary element. Consider any b E Q such

that b S > cl. Then I C c-'bS and since x E IX, by the Definition 2.3.11, x E c-'bS

so that cx E bS. Thus, c c E (ci)". Since x was an arbitrary element, cl" C (cf)'.

Conversely, let x E (CI)". Let b E Q be any element with bS > 1. Then CI C cbS.

Hence, (cl)' Ç cbS, that is, x E cbS. But, b was an arbitrary element. Therefore,

x E n cbS, with bS > 1, that is, x E c (n bS) = ci' and (CI)' C cl' follows.

(4) : Let c E U ( Q ) and x E I-'. Then cIxc-'ci = cIxI C c l implies xc-' E

( C I ) Thus, 1-'c-' E (CI)-'. Conversely, x E (CI)-', implies c l x c i C cl. Hence,

i x c l C 1, that is, xc E Id' . Thus, x E I-L~-L and (ci)-' 1-'c-l follows.

cl The proof of the following result can be found in [MMUSC].

Proposition 2.3.13 Let R be a Dubrovin vaiuation ring of Q and let A be an

R-ideal of Q. Set S = O,(A) and T = Or( i l ) . Then:

(2) A" = A* and (.il-')' = A-'

( 3 ) I/ -4 is not principal a s a right S-ideal then -4-'A = J ( S ) and J ( S ) is not a

principal right S-idenl.

(4) If -4 c A*, then A' = CS and A = cJ(S) for some regular element c E A*. In

particular, A = A'J(S).

An R-ideal -4 of Q is called divisorid if A = A'. With D ( R ) we denote the set

of al1 divisorial R-ideals of Q. On the set D(R) we define the partial operation "O"

by A O B = (AB)' if A, B E D ( R ) with O,(A) = O1(B) . The next result shows that

( D ( R ) , O) is an algebraic structure known as Brandt groupoid without connectivity

property.

Theorem 2.3.14 ( [Dub84]) Let R be a Dubrovin va[uation ring of Q. Then

( D ( R ) , O) has the following properties:

(1) For every A E D(R), there exist unique elernents &(A) and E f ( A ) in D(R) such

that

( 2 ) For A, B E D ( R ) , the product A O B is defined i f a n d only if Er(A) = Ei(B) .

(3) Iffor A, B, C E D(R) the products Ao B and B o C are defined, then (-4 O B ) O C

and -4 O ( B O C ) are dejîned and they are equal.

( 4 ) For every element -4 E D ( R ) t h e m ezists a unique e lement A E D(R) such tha t

Proof. (1) : Let A E D(R) and set & ( A ) := O,(A), E l ( A ) := Or(A) . Then:

&(A) E D(R) : First, we prove that S := O,(Er(-4)) = O,(-4). For q E S, we

have q E O,(A)q C O,(A). Conversely, q E Or ( A ) implies Or(A)q C Or( A), i.e.,

q E S. Hence,

that is, Er(A) = &(A)' E D(R).

0 A O E,(A) is defined since O,(-4) = Oi(O,(-4)) = Oi(E,(A)).

simiiarly, E I ( A ) E D(R), Ei(A) O A is defined and E l ( A ) o A = -4. The uniqueness

of the right and the left units follows by a familiar argument.

(2) : This follotvs immediately from the defini tion.

(3) : Let A, B, C E D(R) and let Ao B and BoC are defined, Le., 0,(.4) = Oi(B)

and Or(B) = Ol(C). It is enough to prove that

((-4B)'C)' = (ABC)' = (.4(BC)')'

If .4B = (AB)' , then the first equality holds. Let AB c (AB)' and let S = O,(AB).

Then S > O,(B) and since Oi(B) = O,(A) > A-'A, we have A-'AB C B. If

A-'AB = B, then O,(B) > Or(-4B) = S , i.e., S = Or(B) . If A-'AB c B, then

-4-'-4 C Ol(B) = O,(-4). By Lemma 2.3.10, this implies that A is not principal

as a right O@)-ideal. Hence, by Proposition 2.3.13 ( 3 ) , A-'A = J(O,(A)) =

J(Or(l3))- Hence, B > J ( O I ( B ) ) B and by the left version of Lemma 2.3.10, B is

principd as a left Oi(B)-ideai. So, B = Oi(B)b ; for some b E B. Therefore, S =

O,(AB) = O,(AOl(B)b) = O,(AO,(A)b) = Or(Ab) = b-'O,(A)b = b-10i(B)b =

O,(Oi( B ) b) = O,(B). Hence, S = Or(AB) = Or(B). Now, since AB is not divisorial,

by Proposition 23-13 (3) and (4), (AB)' = CS and AB = c J ( S ) , for some c E U(Q),

and g ( S ) is not principal right S-ideal. Then g ( S ) - ' J ( S ) C O , ( z ( S ) ) = S by

Lemma 23-10. Hence, S C Z(S)- ' implies J ( S ) = S J ( S ) 2 g ( S ) - ' J ( S ) 2 J ( S ) ,

i.e., J ( S ) - ' J ( S ) = J ( S ) . This irnplies J(S)- ' C O r ( g ( S ) ) = S , i.e.. J(S)- ' = S.

Finally, by Propositions 2.3.12 and 2.3.13, since Or(C) = O,(B) = S and J ( S ) =

J(Oi(C)) = C we have (ABC)' = (cZ(S)Cjx = c(Z(S)C)^ = c(C)' = (CC)' =

(cOi(C)C)* = (CSC)' = ((AB)'C)'. Similady, (ABC)' = (A(l3C)')'.

(4) : Let A E D(R) and denote S = O,(A), T = Of(A-'). Then S = T. To

prove this, first note that R and S are overrings of R. Since by Theorem 2.3.3 and

Proposition 2.3.2, overrings of R are linearly ordered by inclusion, we can assume

that R S C T.

If A-'A = S, then by Lemma 2.3.10, A = a s for some regular element a E -4 and

also A-' = Sa-'. Hence, q E T implies qSa-' C Sa-', i-e., q E S. So, T C S.

If A-' A c S , then K L A C J ( S ) . Hence, 1 + A-% C 1 + 3 ( S ) C U ( S ) c U ( T ) ,

i-e., A-'A 2 J ( T ) . Assume that S c T. Then A c AT, since otherwise -4 = AT

and T C O,(A) = S. By Lemma 2.3.1, there exists a regular element c E A T such

that A C c J ( S ) C CS C -4TS C .4TT = AT. Multiplying by T on the right, we have

AT = cT. Furthermore, A 5 CS implies c-' E .a-'. Thus, T = c-'-4T A-'AT C

J(T)T = J ( T ) , a contradiction that shows that S = T, i-e., O,(A) = 01(.4-').

Similarly, O, (A-' ) = Ol (A) .

This impiies that the products A-'oA and Ao A-' are defined and by the definition (1) A-' O A = (A-'A)' Ç S* = O,(A)' = O,(A) = S. Hence, A-' o A E. O,(A).

Let c E U ( Q ) be any element such that A%4 C CS. Then c- 'X 'A C S, Le.,

c-'A-' 2 A-'. Hence, C-l E OL(;2-') = O,(A), i.e., 1 E CS. This implies S C

n CS = ( A-' A)' = A-' o A, i.e., A-' O -4 = O,(A) . Similarly, -4 o A-' = OI(A). cS2.44 A

O

Now, we consider a Dubrovin valuation ring R of rank one, that is, J ( R ) and (0)

are the only Goldie prime ideals. Ln this case, the only proper overring of R is Q, and

the opetation "O" is defined on the whole of D ( R ) . The following result shows that for

a rank one Dubrovin valuation ring R, the structure ( D ( R ) , O ) from Theorem 23-14

becomes a group.

Theorem 2.3.15 Let R be a Dubrouin ualuation ring of Q of rank one. Then

( D ( R ) , o ) is a group.

Proof. By Theorem 2.3.3 t here are no proper overrings of R in Q. Hence, for every

A E D(R), O,(A) = Ol(A) = R. Thus, for any two elements A? B E D(R),

O,(A) = Oi(B) = R, that is, A O B is defined. So, "O" is defined on the whole of

D(R)-

Furthemore, there exists a unique element E = R E D ( R ) such that for every

A E D(R) , A O E = E O A = A, that is, there is an identity element in D ( R ) .

Finally, for every A E D(R) , there is a unique element A-' E D( R) (see the proof

of Theorem 23-14) such that -4 O -4-L = Or(A) = O,(-4) = A-' O -4 = R = E. Since

"0" is associative, ( D ( R ) , O ) is a group.

O

Now, we consider Goldie prime ideals of a Dubrovin valuation ring. We first show

that in the special case of the total valuation ring, the classes of Goldie prime ideals

and completely prime ideds coincide.

Proposition 2.3.16 Let R be a total ualuation r ing in a skew field Q , and let

P be a prime ideal of R. Then, P is a Goidie prime if and only if P is completely

prime.

Proof. Assume that P is a Goldie prime. Then, by Theorem 2.3.3, S = Rp zp R

is a Dubrovin valuation ring of Q and J(S) n R = P. But, by Corollary 2.2.5,

J ( S ) 2 J ( R ) c R, so that P = J(S). On the other hand- S as an overring of

R in Q, is a total valuation ring in Q. Hence, Ç = S/J(S) is a skew field. Thus,

R = R / P = R / J ( S ) is a subring of a skem field S, that is, R / P is an integral

domain. So, P is a completely prime ideal.

Conversely, if P is completely prime ideal, then R/ P is an integral domain. Also,

R is a Dubrovin valuation ring in Q with R = R/J(R) is a skew field. Hence, d(R) =

d ( R / J ( R ) ) = 1. By Theorem 22.8, R is a l-chain domain; so by Remark 1.1.11,

E = R/ P is also 1-chah dornain such that R / z ( E ) = RI P / T ( R ) / P "- R / z ( R ) is

a skew field. Again, by Theorem 22.8, R / P is a Dubrovin valuation ring in R, Le.,

R / P is a prime Goldie ring. Thus, P is a Goldie prime.

T h e natural question is: Does there exist a Dubrovin valuation ring with a prime

ideal that is not Goldie prime? This question was raised by H.H.Brungs and G.Torner

in [BT76]. in the form: Does there exist a prime c h a h ring with zero divisors, Le., a

chain ring with a prime ideal that is not completely prime? N-Dubrovin, [Dub93],

gave the positive answer t o this question.

Example 2.3.17 ( [Dub93]) There exists a Dubrovin ualuation ring with a

p r i m e ideal that is not Goldie prime.

Proof. Consider the group SL(2 , R ) of 2 x 3 matrices with real entries and determi-

nant 1. Then

are subgroups of SL(2 , R ) and every element s E S L ( 2 , IR) can be written in a unique

way a s s = r ( p ) u for r ( y ) f S, O 5 y < 27r , u f U.

Consider now first the universal covering group l? of S where l? = {xt 1 t f R } with

xt1xt2 = xtl+t2 as operation, and xt' 5 xfz if and only if t 5 t 2 . AS a linearly ordered

group, I' is isomorphic to (B, +, 5). The covering map r from R t o S with r(xt) = r ( t )

is a homomorphisrn with Ker T = (x2"), the cyclic subgroup of B generated by x2'.

Next we define the covering group G of SL(2 , B) as the set G = { x t u 1 xt E r, u E

LJ} with the operation given by:

where 9 E [O, 2rrj and u l r ( y ) = r(+)u; for $ E [O, 2 ~ ) . The mapping T above is

extended to a map from G to SL(2 , R) by defining r (x t i r ) = r ( t )u .

Finally, we consider the subsemigroup P = {xru 1 O 5 r E R, u E U) of the group

G. Then P U P-' = G and P n P-' = U. The element x" is contained in P and is

central in P. So, Q = x"P is an ideal of P, properly contained in the maximal ideal

J ( P ) and, in addition, n Qm = n xTmP = 0. Since x"/2xr/2 E Q but x " / ~ Q, Q is

a prime ideal in P that is not completely prime. It can be shown that P satisfies the

conditions of Theorem 1.3.3. hence, there exists a skew field D and a total valuation

ring S in D associated to the cone P. By Theorem 1.3.1, the corresponding ideal

y(Q) = QS = xTPS is a prime ideal of the ring S that is not completely prime. Now,

applying Theorem 1.3.5 and Proposition 2-3-16, S is a Dubrovin valuation ring with

a prime ideal y(Q) that is not Goldie prime.

17

The following results show that Goldie prime ideals of a Dubrovin valuation ring

R in Q have nice properties. These resuIts are important steps for describing the

structure of ideals in Dubrovin valuation rings. But first, we prove the following

basic fact which we will use frequently.

Lemma 2.3.18 L e t R be a Dubrouin uafuation ring of a simple artinian ring Q

and let S be an ouerring of R in Q. If 1 # R is a non-zero ideal of R sach that

I 3 g ( S ) , then iS = S.

- Proof. The ideal I/J(S) is a nonzero ideal in R = R / g ( S ) which is a Dubrovin

valuation ring in 3 = S / g ( S ) by Theorern 2.3.4. Since R is a prime Goldie ring, the

ideal I / J ( S ) is essential and by Theorem 1.1.5, it is a regular ideaI. So, there exists - an element r E 1 so that r + J(S) is a regular element in R = R / J ( S ) . Hence,

r + J ( S ) is a unit in 3 = S / g ( S ) , that is, r E U(S). Thus IS = S.

Proposition 2.3.19 Let Pi be Goldie primes in R, i E A. Then P = n Pi is a

Goldie prime.

Proof. Since Pi is Goldie, the localkation Rp, eXists for every i and we set S = U Rp,.

The overrings Rpi of R in Q are totally ordered by inclusion. Hence, S is a ring in Q

containing R. By Theorem 2.3.4, S is a Dubrovin valuation ring of Q and z ( S ) C Pi

since Rpi C S. It follows that z ( S ) is a Goldie prime contained in P (Theorem 2.3.3).

Assume that P > z ( S ) . Then by Lemma 2.3.18 we have PS = S. Hence,

1 = C pisi for elements pi in P and si in S = U Rpi. But, the rings Rp, are totally

ordered by inclusion. SO, there exists an index jo E A with si E Rp,, for al1 i and

pi E P C Pjo-

Therefore, 1 = C p i s ; E Pj0Rp,. = Pjo, a contradiction that shows P = J(S),

which is a Goldie prime.

O

Lemma 2.3.20 Let I # R be a non-zero ideal in R with I = 12. Shen I is

neither a principal right O,(I)-ideal nor a principal left O[([)-ideal.

Proof. Both S = O,(I) and T = Oi(I) are overrings of R and hence either T C S

or S C T? since the overrings of R are linearly ordered by inclusion. It is enough

to consider the case T C S. We show first that I 2 J ( S ) . Otherwise, I > J ( S )

and by Lemma 2-3.18, it follows that S = IS = I, a contradiction that proves

I J ( S ) . Assume that I is principal as a right S-ideal, i-e., I = as. Then, by

Corollary 1.1.4, a is regular element. Then, a S = I = IZ = aSaS implies S = Sas

and 1 = Cy='=, siat;, si, t ; E S, follows. Since S = Rs(s) = J(s$3, t here exist elements

c. d in CR(J(S)) with CS; E R, t id E R for al1 i. Hence cd = C cs;at;d E I Ç, g ( S ) ,

a contradiction which proves that I is not a principal right S-ideal. Since T 2 S

implies I 5 J(S) 5 J ( T ) , a similar argument shows that I is not a principal left

T-ideal.

O

The next result shows that idempotent ideals # R are Goldie primes.

43

Proposition 2.3.21 Let I2 = I # R be an idempotent ideal. T h e n the following

hold:

b) 1 = J ( S ) is a Goldie prime with S = R3(sp

Proof. Let S = O,(I) and T = Or([). It is enough to consider the case S C T. From

Lemma 2-3-20, it follows that I is neither a principal right S-ideal nor a principal left

T-ideal. Hence, I-'1 = J ( S ) and II-' = J ( T ) by Proposition 2.3.13(3). Further,

Id' = {x E Q 1 X I C S } = ( S : I ) [ _> T. Conversely, if x E (S : I ) ( , then X I C: S

and XI = x12 C S I C I , and x E T follows; we have proved that 1-' = T. However,

IIdi = J ( T ) C g ( S ) J ( R ) c R implies 1-' C (R : I),. Further, if q E ( R : I),,

then Iq & R C O,(I), i.e., Iq = I I q C I . Hence, q E O , ( I ) = S , i.e., ( R : I ) , C S.

Thus T = 1-' C S and T = S follows which proves a).

Now, 37s) = 1-'1 = T I = I which proves that I is Goldie prime, since J ( S ) is

a Goldie prime. In addition, S = R3(s) follows and both parts of b) are proven.

O

The nest result shows that the union of Goldie primes is again a Goldie prime.

Corollary 2.3.22 Let R be a Dubrovin ualuation ring and let R > Pi, i E h, be

Goldie primes in R. Then:

a) P = (J Pi is Goldie prime;

Proof. If there exists a Pj with Pj > Pi for al1 i, then P = Pj is a Goldie prime,

Rp = Rp, = n Rps, and P = z( R p ) implies Q ( P ) = O, ( P ) = R p , by Lemrna 2.3.8.

We can therefore assume that for every Pi there exists a Pj with Pj > Pi::. Hence,

P > Pi for al1 i, and P > P2 > Pi for all i, since otherwise Pz Pi, which would

irnply P C Pi, since Pi is prime. It foliows from Proposition 2.321 that P = P2

is a Goldie prime with Rp = O I ( P ) = Or(P)- It remains to prove that S = O!(P)

where S := n Rp, . Let x E O l ( P ) , hence x P C P. Since P > Pi is an ideal in R

and R/P, is Goldie, P contains an element in CR(Pi) and PRF. = Rpi. Therefore,

xRe = x P R R c PRp, = Rpi ; so x E Rpi for al1 i; hence x E S follows. Conversely, if

x E S and a E P, then there exists Pi with a E Pj, and xa E SPj C RpJPj = P, c P

proves x E O[( P ) ; so S = Or ( P ) follows.

O

Now, let I # R be an ideal of R that is not Goldie prime. Then, the families:

R = {P 1 P is a Goldie prime, I c P}

S = { P 1 P is a Goldie prime, P c I }

are not empty since J ( R ) E R and (0) E S. By Proposition 3-3-19 and Corol-

lary 23-22, Pl = n { P 1 P E R) and P2 = U { P 1 P E S ) are Goldie primes,

Pl > 1 > PZ and and no further Goldie prime ideal exists between Pl and R2.

Definition 2-3.23 Let R be a Dubrouin ualuation ring of Q . A p r i m e segment

of R (and of Q ) is a pair of two distinct Goldie primes Pl > P2 > ( O ) in R so that

no further Goldie prime exists between Pl and Pz.

The next result is the most important for the study of prime segments of a

Dubrovin valuation ring.

Theorern 2.3.24 Let I # R be an ideal in the Dubrovin valuation ring R. Then

r) In = P is Goldie prime.

Proof. The result follows if 0 In = f m for a certain m, since then ( I m ) 2 = Im is

idempotent and we can apply Proposition 23.21. We c m therefore assume that

and show that the assuming P not to be Goldie prime leads to a contradiction.

First, let A and B be ideals of R such that P C *4 and P c B. Then there exists

an 72 such that In C A and 1" B. Otherwise, for al1 n, A 2 In c 1 or B C In C 1,

Le., -4 C n In = P or B C In = P, a contradiction. Thus, .AB > I n l n = 12" > P;

so, P is a prime ideal.

If 1 itself is not Goldie prime, then by the remark above, there exists a prime

segment PL 2 P2 in R with Pl > I I ) P2.

If I itself is a Goldie prime that does not have a lower neighbor among Goldie

primes, then I = (J Pi for Goldie primes I > Pi. In this case, I > I2 > Pi for al1

i and hence I = 12. So, P = n In = I is a Goldie prime, which is a contradiction,

since P is assumed not to be Goldie prime.

Therefore, we can assume that there exists a prime segment Pl > P2 in R with

Pl > I 2 P2 - We define N : = Pi 1 Pl C 1. Then I3 = I l i E Pl I Pl = iV and t herefore

n In = n N n = P follows; in addition, N and P are Rpl-ideals. After localizing at

Pi we obtain Rpl > Pl > N > n Nn = P > PZ and P is not Goldie prime in Rpl.

We therefore can consider Rp, / P2 and can assume from now on that R has rank one

with R 3 J ( R ) = P, 2 1V II n N n = P 3 (0).

We consider the following set W of ideals in R :

W = { L 1 Pl > L > P, L an ideal of R);

note that W contains Nn for n 2 2. Two cases may happen.

Case 1- : IV contains a n ideal L which i s not divisorial

Then L c Lœ where L' = n C R with CR > L (by Definition 2.3.11 and the fact

that R is of rank one and hence O,(L) = R).

It follows from Proposition 2.3.13 (1) and (4), that L' = (15-')-', L' = a R and

L = aJ(R) = aPL for some unit a in Q. Note that a is regular but not a unit in R

sinceotherwise L = a J ( R ) = a R g ( R ) = R J ( R ) = J ( R ) = Pl. Aiso, L* = a R C Pl.

Since O I ( L = ) = aRa-l = R we have Lm = a R = Ra and (L-)" = anR = Ran for

n 2 1.

It follows that the set C = {an 1 n = 1,2,. . . ) is an Ore system in R, and since

a-' 6 R, we have R c RC-' C Q ; Le., RC-l is a proper overring of R. Hence,

RC-l = Q. Since P is a non-zero ideal in R, it contains a regular element c and

C-l = ra-" for some r in R and some n 2 1. Hence, an = cr E P: which implies

(L')" = anR 5 P ; so Lw 2 P c L c Lm7 which is a contradiction.

Case 2. : L = L u f o r aLZ L E t V

Let L E W and consider the R-ideal L-l ; it is divisorial since (L-')= = ((L-l)-l)-l =

(L=)-l = L-l. We cclairn that L-' > B. Otherwise, L-' = R and (L-IL) ' = L* = L.

On the other hand, by Theorem 2.3.15, (L-' L)' = L-' O L = Or(L) = R, a contra-

diction.

We consider ilo = (J L-l? L E W, and want to prove that A. is an overring of R,

hence equai to Q. Let x, y be elements in A. and x E L;', y E LyL for LI , L2 f W

follows. Either Ll C L2 or Lz C L L and we can assume LI C L S -

Since

for any non-zero ideal L of R, it follows that L;' C L;' and s & y E L;'.

Further, L;' LY 'L~L* 2 L;'RLl C L;' L~ C R shows that LF'L;' C

(R : L2L1)i = (&Li)-', and xy E (L2Ll)-l. In order to see that xy E Ao, note

that Pl > Li > P and Pl > Lz > P imply Pl > Lz Ll > P since P = n Nn and

Nn+l C Nn for al1 n.

To reach the final contradiction we choose a r e g d u element c in P # (O). Since

Q = AO = U Lw', L E W, there exists L in W with c-' E L-'. Therefore, C-'L C

L-'L R and L E CR C P7 but P c L. It follows that P = n In is Goldie prime.

Lemma 2.3.25 Let R be a rank one Dubrouin valvation r ing . Then the group

( D ( R ) , o ) is order isomorphic tu a subgroup of (R, +), the additive group of real

nurnbers.

Proof. CVe define a binary relation " 2 " on D( R) by A 2 B if and only if -4 C B,

for all A, B E D(R). Clearly, the relation " 2 " is reflexive, antisymmetric and

transitive, Le., it is a partial order on D(R). Since the elements in D(R) are totally

ordered by incIusion, the relation " 2 " is a total order. Furthermore, -4 C B

implies AC BC and CA C C B for al1 A, B, C E D ( R ) . Hence, A C B implies

-4 O C = (-AC)' C (BC)' = B O C and C O A = (CA)' (CB)' = C O B for al1

A, B, C E D(R). Therefore. " 2 " is compatible with " O " and (D(R), O , 2) is a

totally ordered group.

Vie prove that D(R) is an archimedean group, i-e., for a11 1, B E D(R) with

1 C R, there exists an integer n such that ( In)* c B.

Clearly, it is enough to prove that for every I E D( R), I C R , the intersection

ri = n ( r n ) - = (01. Let 1 E D(R), I c R. Then I C J = J ( R ) . If I c J, then by Theorem 2-3-24,

n in = ( O ) , since J > (O) is a prime segment. If I = J > J2, then by the same

argument, again In = (O) . We are left with the case I = J = J2. In this case, by

Lemma 2.3.10, J is not principal as a right R-ideal, since O,(J) = R. This implies

that J' = R. In fact, if J C CR for some c E U(Q), then c-'J C R and, hence,

Rc-'RJ 2 R. If Rc-'RJ = R, then ( R : J)iJ = R, i.e. J-' = (R : J) i . This implies

J-'J = R = O,(J) and by Lemma 2.3.10, J is principal as a right R-ideal, a con-

tradiction. So, Rc-'RJ C R, i-e. Rc-= R J 2 J which implies Rc-'R C Oi(J) = R,

Le. R C CR. This means that I* = J* = R > J = 1 , that is, 1 is not divisorial, a

contradiction. Therefore, the case I = J = J2 cannot arise; so n In = ( O ) holds for

any I f D(R), 1 C R.

Now, assume that K = n(In) ' # (O). First note that I< is divisorial, since

Ic 2 ( In)= for al1 n implies h" C ((In)')' = (In)' for al1 n, i-e., Ii' C n(In)* = Iï

and K = K* follows. Hence, there exists an integer k, such that K > ik. Otherwise,

1; c n Ik C n ( l q x = A', i-e. (O) = 1;. NOW, K' = K > (1")' 3 ( IN1)= , since if

( r k ) = = (Ik+')=, then I = R, because ( D ( R ) , o ) is a group. Hence, K > (Ikf')- > n ( ï k ) - = I<, a contradiction which shows that I< = n ( I n ) ' = (O) , i.e. (D(R), O) is

an archimedean group. By Holders Theorem, (see for example, [Sch50]- Theorem 1,

Chapter l), D(R) is order isomorphic to a subgroup of (W. +).

In the case of a Dubrovin valuation ring R of rank one, the group (D(R), O, 2)

contains a subgroup we denote by H(R) := {I E D(R) 1 I = aR, O # a E Q), i-e.

the set of al1 those non-zero ideals I which are principal as right R-ideals. For,

if 1 = aR E H ( R ) , then a is a regular element, i.e. a is a unit in Q. Hence,

Oi(a R) = a Ra-' and since R is of rank one, aRa-' = R, i.e. a R = Ra. Therefore,

for aR, bR E H(R) we have aR O bR = (aRDR)' = (abR)' = abR E H(R) and

I-' = Ra-' = a-'R E H(R).

Let I be any non-zero R-ideal in Q. Then 1 is either divisorial and then I = I' E

D(R) or I c 1' and then by Proposition 2.3.13 (4), 1' = a R = Ra E D(R) and

I = aJ(R)(R) for some a E U ( Q ) . Therefore, the lattice of al1 ideals of rank one

Dubrovin valuation ring R is known completely if D(R) and H ( R ) are known.

The next result describes two possibilities for an arbitrary p i m e ideal in a Dubrovin

valuation ring R.

Proposition 2.3.26 Let R be a Dubrouin valvation ring of Q , [et P be a prime

ideal of R, and S = O,(P) = Ol(P) . Then P is either a Goldie prime and P =

J ( S ) , o r P is not Goldie p r i m e and J ( S ) is the minimal Goldie pr ime containing

P. iWoreover, i n the second case, there is no ideal of R properly between P and z ( S ) .

Proof. Since 1 + P C U ( R ) C U ( S ) , clearly P C J ( S ) .

If P = J ( S ) ? then P is a Goldie prime, by Theorem 2.3.3.

If P c J ( S ) , then P is not a Goldie prime, since otherwise R p P = P , i.e.,

Rp C O 1 ( P ) = S , which contradicts S c Rp.

Let I be an ideal of R such that P c I C z ( S ) . Then by Theorem 2.3.24,

Pt := n In is a Goldie prime. Also, P c Pl, since P c In, for al1 n. Otherwise,

there exists an n such that I" C P and then I C P, a contradiction. Furthermore,

POr(P1) P' = POl ( P l ) P' C P P' 2 P. Since P O r ( P f ) C P' c R, clearly PO, ( P t ) is

a two sided ideal of R. Hence, PO,(Pf) C P since P is a prime ideal and P' P.

so, O r ( P f ) C O,(P) =S.

Hence, by Lemma 2.3.8,

Corollary 2.3.27 If P is a prime ideal of a Dubrovin valuation ring R that is

not Goldie prime and S = O,(£') = Oi(P) , then J ( S ) > P > n Pn is the "highest"

prime segment i n ring S .

Chapter 3

Prime Segments

The existence of a Dubrovin valuation ring R in a simple artinian ring Q wi th a prime

ideal I that is not Goldie prime irnplies the existence of a prime segment Pl > I > Pz,

consisting of a pair of two distinct Goldie primes Pl and Pz in R such that no further

Goldie prime esists between Pl and Pz (see Example 2.3.17 and the remark after

Corollary 2-3-22).

In the last decade, prime segments were defined and studied in different situations.

In [BD], H.H. Brungs and N.I. Dubrovin considered prime segments in total valuation

rings . They classified prime segments and in the rank one case described the structure

of al1 ideals. Also, examples for al1 cases were constructed. In [BS95], H.H. Brungs

and M. Schroder defined prime segments in valued skew fields and gave methods for

constructing valued skew fields with prescribed types of prime segments. In [DD96],

the structure of ideals and prime ideals in a cone P of a right ordered group G is

studied. The rank one case is described completely and corresponding examples are

constructed. In [BTgS], H.H. Brungs and G. Torner defined right cones and classified

t heir prime segments.

Note that prime segments correspond to jumps in ordered and right ordered

groups. Let (G, P ) be a right ordered group with the positive cone P, Le., PP C P ,

P n P-' = { e } and P U P-l = G. A subset H C G is called convex if x, y E H

g E G, x <_ g 5 y implies g E H. The set Z of al1 convex subgroups of G is a

chain containing ( e ) and G. If CA (A E h ) is in S, then n C.1 and U C.1 are again

in E. Hence, every element g # e in G defines two subgroups Dl C E C so tha t

C > Dl g E C \ D, and C contains no convex subgroups strictly between C and D.

Such a pair C > D is called a jump in C (see [Iiop98] and [Fuc63]). In this situation

there exists a 1 - 1 correspondence between Z and the set of al1 completely prime

ideals of the cone P. More preciselÿ:

Theorem 3.0.28 Let (G, P ) be a right ordered group.

If A E Co A # G, is a conuex subgroup of G then I = P\ ( P il) is a completely

prime ideal of the cone P.

Conuersely, if 1 # P is a comp[etely prime ideal of the cone P , then A = ( P \ 1) U ( P \ 1)-l is a conuex subgroup of G.

By this correspondence, jurnps in C correspond to the prime segments in the cone

P.

In this chapter, prime segments in a Dubrovin valuation ring R are classified and

the structure of the lattice of al1 ideals in the rank one case is completely described.

The results obtained show that there exists a complete analogy between the ideal

structure of a cone in a right-ordered group and the structure of ideals of a Dubrovin

valuation ring in a simple artinian ring.

3.1 Prime segments in Dubrovin valuation rings

In this section, R denotes a Dubrovin valuation ring in a simple art inian ring Q.

Definition 3.1.1 Let Pl ZI P2 be a pr ime segment of R.

52

The prime segment Pl > P2 is called archimedean if for al1 a E Pl \ P2 there exists

an deal I E Pl such that a E I and n In = Pz.

The prime segment Pl > Pz is- called simpIe i f there are no ideals between Pl and

p2

The is called exceptional i f there exists a prime ideal I in R that is not Goldie

prime such that Pl 2 i 3 Pz-

Lemma 3.1.2 Let Pl > P2 be a prime segment in R.

i) If PL > P2 W exceptional and i is a prime ideal that is not Goldie prime such

that Pi > 1 3 Pz, then there are no ideais between Pl and I , Pl = P: and

nrn = pz,-

ii) If Pl > P: or Pl = U 1, I c Pl, Le., Pl is the union ofideals 1 properly contained

in Pl, then the prime seg.ment PL > Pz is archfmedean.

Proof. i) Let L be an ideal of R such that PI > L 2 1. Then by Theorem 23.24,

P = Ln is a Goldie prime ideal of R and Pl > L > P 3 1 > P2, a contradiction.

Furthermore, P: > I since otherwise P: Ç 1, which implies Pl C 1. Hence, Pl = P:.

A ~ S O , n rn = pz.

ii) First, assume that Pl > P: and let a E Pl \ Pz. Set 1 := Pl. Then a E I and

n In = n PT C PL c Pl Hence, n In = P2 since n In is a Goldie prime.

Next, assume that Pl = (J 1, the union of al1 ideals I properly contained in Pl

and let a E Pl \ Pz. Then there exists an ideal 1 c Pl such that a E I. Then

Pl > I > n In 2 Pz, i-e n in = PZ, by the same argument as in the first case.

O

The next result shows that there are exactly three types of prime segments in a

Dubrovin valuation ring R.

Theorem 3.1.3 For a prime segment Pl > f i of a Dubrouin uahation ring R

exactly one of the follo-wing possibilities occurs:

a ) Pl > Pz is archimedean;

b ) Pl > P2 is simple;

c ) Pl 3 P2 is exceptional.

Proof. Let L := (J 1 be the union of al1 ideals 1 of R properly contained in Pl. If

L = P2, t hen there are no ideals between Pl and Pz, Le., the prime segment Pl 3 P2

is simple.

Next, we prove that the prime segment Pl 3 P2 is exceptional if and only if

Pl > L > P2 and Pi = P:.

Let these conditions be satisfied. Then L is a prime ideal of R. For, if B and C

are ideals of R such that B > L and C > L then B 2 Pl and C _> PL since otherivise

B C Pl or C C Pl, Le., B C L or C C L. Hence, BC > P: = Pl 3 L. So, L is a

prime ideal that is not Goldie, i-e., the prime segment PL > Pz is exceptional. The

converse was proved in Lemma 3.1.2.

We are left ivith the case that Pl > P: or Pl = U 1 for ideals I of R with

Pl > 1 > P2. In both of t hese cases the prime segment Pl 3 P2 is archimedean, as

we have shown in Lemma 3 - 1 2

cl

Consider the set K ( P 1 ) := {a E Pi ( PlaPl c Pl) . Theo, for any r E R and

a E I i ( P l ) , PlruPl Ç PlaPl c Pl and PlarPl C PinPl C Pt, i.e., ra, ar E I i ' (9 ) .

.-lso, for a , b E IiW(Pl) we have PlaP1 E PlbPl or PlbPl C PlaPl. In the first case,

Pl ( a + b) Pl C Pl bPl c Pl. Hence, a + b E Ir'(Pl). Similarly, in the second case,

a f b E Iip(Pl). Therefore, Ii(Pl) is an ideal of R. Using the ideal K( Pl ), we can

prove the following result :

Corollary 3.1.4 The prime segment Pi 3 Pz of R is archimedean if and only i f

IC(Pl) = Pl, it is simple i f and onZy i f I i (P l ) = P2 and it is exceptional if and only

if PL > K ( P l ) 3 P2. In the fast case, K ( P 1 ) is n prime ideal that is not Goldie.

Proof. First, let Pl > P2 be archimedean. For any a E Pz, PlaPl C - Pz c PL, i.e.,

a E K(Pl ) . If a E Pl \ P2 then there exists an ideal 1 c Pl, such that a E 1. Hence,

P1aPL C I c Pl, Le., a E I i (P l ) . So, Pl C K ( P l ) C Pi, Le., l i ( P l ) = PL.

Conversely, let K ( P l ) = Pl and assume that Pl > Pz is not archimedean. If

Pl > P2 is simple and a E Pl \ P2 then f i > P& > &, a contradiction. If PL > P2

is exceptional, then for a prime ideal I that is not Goldie such that PL > I > P2 and

a E Pi \ I , we have Pla PL = Pl Ra RPl C 1 c PL, since there are no ideals between

Pl and 1 by Lemma 3.1.2 i). But 1 is a prime ideal of R. Bence, PL C 1 or Ra R C 1,

a contradiction which shows that Pi 3 Pz is archimedean.

Next, let PL > 6 be simple and assume that there exists an element a E I i ( P l ) \ P2. Then PL > PluPl 2 P2, since PLaPl = Pl RaRPl 2 P2 implies Pl 2 P2 or a E P2-

But this is impossible. So, l i ( P l ) = fi.

Conversely, let K ( P l ) = P2 and assume that Pl 2 P2 is not simple. If Pl > Pz

is archimedean, then for a E Pl \ Pz there exists an ideal 1 c PL with a E 1. Then

PlaPl 2 I c Pl, Le., a E I i (P l ) = P2, a contradiction. If Pl > P2 is exceptional

than for a prime ideal I of R that is not Goldie with PL > 1 3 P2 and a E I \ P2, we

have fiaPl C I c Pl, i.e-, a E L'(Pl) = P2: a contradiction.

Finally, let PL 2 P2 be exceptional and let I be a prime ideal of R that is not

Goldie such that PL > I > Pz. If a E IC( Pl) \ I then Pla Pl C 1 C Pl . Hence, Pl C I

or a E 1, a contradiction which shows that I = K ( P l ) .

Conversely, let Pl > K(Pl ) > Pz. First, there are no ideals of R between PL and

h'(Pl). For, if 1 was an ideal of R such that Pi > I > I i (P l ) , then for a E I\ K ( P l )

ive would have PLa Pl E 1 c PL, Le., a E &.'.(Pl). Secondly, K ( P 1 ) is a prime ideal of

R. For, if B and C are ideals of R such that B > K ( P l ) and C > IC(Pl), then, as

noted first, B > Pl and C > f i . But Pl = P:, since P: c Pl implies P; c Pl, Le.,

Pl C I i (Pl) , a contradiction. Hence, BC > P: = Pl > K(P1). So, Ii(P1) is a prime

ideal of R that is not Goldie, and the prime segment Pl > P2 is exceptional.

It follows from this result that the type of the prime segment Pl > Pz is the same

for any Dubrovin valuation ring R of Q that contains this prime segment.

3.2 Rank one and discrete Dubrovin valuation rings

Let R be a rank one Dubrovin valuation ring of Q. Then Q is the only proper overring

of R and we have shown, see Theorem 2-3-15 and Lemma 2.3.25, that (D(R), O, 2) is

a group, order isomorphic to a subgroup of (R, +). Also D ( R ) contains the subgroup

H ( R ) of al1 nonzero ideals which are principal as right ideals. Using these facts and

Theorem 3.1.3, we now proceed to give a complete description of the lattice of two

sided ideals for rank one Dubrovin valuation rings.

Theorern 3.2.1 Le t R be a rank one Dubrouin valuation r ing o f the simple ar-

t in ian ,ring Q with m a x i m a l ideal J = J(R).

Then exactly one o f t he following possibilities occurs:

a ) The segment J > ( O ) is archimedean and either

i ) J > J 2 and t h e n D ( R ) ( J ) H(R) is an in f in i te cyclic group; o r

ii) J = J 2 and t h e n D(R) i (R, +) and H ( R ) is a dense subgroup of D(R).

b) T h e segment J > ( O ) is s imple and then D(R) = H(R) = {R) is the trivial group.

c ) T h e segment J > ( O ) is exceptional. In th i s case i f 1 is a n o n Goldie pr ime of

R with J > I > (O) , then D(R) = (1) is the inJinite cyclic group generuted b y

I = 1' and a n integer k 2 O ez is ts with H ( R ) = ((Ik)') ( the segment is said t o

be exceptional o f type b).

Proof. We saw in Lemrna 2.3.25 that D ( R ) is an archimedean group.

C L A I M : A s s u m e that R contains a max imal divisorial ideal 1 C R, and let C C R

be a n y divison-al ideai. T h e n there exists a n integer n 2 1 such tha t C = (In) ' .

Since ( D ( R ) , O ) is archimedean with the identity element R, there exists a minimal

n with n 3 1 and C > In. Hence In-' > C > (In)' and (In-')' > C. Therefore,

since " O '' is compatible with "2" we have

and, by the maximality of 1, since C O (I-("-'))= = (C(I-(n-l))-)* is divisorial, we

have I = C 0 (I-("-'))'. Therefore, C = (In)' and the clairn is proved.

a) Let J > (O) be archimedean. Two cases occur, J > J2 and J = J 2 .

By Lemma 23-10, we have J > J2 if and only if J is principal as a right R-ideal,

since S = O,( J ) = R, Le., J = a R. Then, as noted after the proof of Lemma 2-3-25?

O i ( J ) = aRa-' = R. Hence, J = a R = Ra C R is a maxima1 divisorial ideal of R.

By the claim, J is a genrrator of the group D(R); so this proves the case a) i).

Next we consider the case a) ii) where J = J 2 and J > (O) is an archimedean

segment. For every non-zero element a in J there exists therefore an ideal Il 2 J

such that a E 11 and n 1; = O. Then RaR C Il c J, since Ii = J would imply

J = n Jn = n 1; = O, which is a contradiction. We want to show that the ideal

I = RaR is a principal right R-ideal for any O # a in J and hence that I E H ( R ) .

If 1 is not right principal, then IJ = I (Lemma 3-3-10) and a = CQ, riasi with

ri E R, S i E J , for al1 i. Since R is a left Bezout order there exists s in R with

Rsl + - - - + Rs, = Rs; hence O # s E J, T2 = RsR c J and 1 = IT2 follows.

Since Tl = J and O,(T*) = R is right Bezout, we can apply the left-right sym-

metric version of Lemma 2.3.1 to obtain a regular element to in TL = J such that

T' = RsR C Jto. Hence, I = TT2 2 IJto C I J = I and I = I J t o = Tto foliows. Since

to is regular, f o l exists in Q. Now, Itol = 1; so tôl E O,([) = R, a contradiction

since to f J.

This proves that 1 = RaR is in H(R) for any O # a E J. Finally, for every

RaR c J there exist b E J\ RaR and RaR c RbR c J follows; H(R) and D(R) are

therefore isomorphic to dense subgroups of (R, +). Since the intersection Ii = n Ii of

divisorial ideals of R is divisorial if I< # (O), D(R) is also complete and D(R) - (Et, +)

follows.

The assertion in case b) is trivial.

It remains to consider the case c) where J > I > ( O ) is an exceptional prime

segment. In this case, J = J2 is not principal as a right R-ideal, and as in the

proof of Lemma 2.325, Js = R, i.e J is not divisorial. But I' = I , since otherwise

I 3 1, J > 1, yet 1 = I*J by Proposition 23 -13 (4), which is a contradiction for

the prime ideal 1. Hence, 1 is a maximal divisorial ideal (there are no ideals of R

between J and 1). Therefore, D(R) = (1) by the claim and H ( R ) is then equal to

((1")-) for some li 2 0.

O

The exceptional case is the most interesting. It splits into countably many sub-

cases, depending on the integer li 2 O. We give chains of d l ideals in every case.

If R = O, then H ( R ) = {R), i.e., there are no principal right ideals. If L is any

ideal of R, then either L = L' E D(R) = (1) or L c L* and then L' is a principal

right ideal of R, i.e.: Lx E H ( R ) . Hence, L' = R and L = J. Therefore, the proper

ideals of R are J, (0) and powers of 1, i.e,

is the chain of al1 ideals of R.

If k = 1, then H ( R ) = (1) = D(R) . Hence, 1* = 1 = aR = Ra is principal,

(In)*=aRoaRo---oaR=(aRaR-.-aR)'=anR= Ran and

is the chain of al1 ideals of R.

Ifk > 1, then H ( R ) = ( ( I V * ) ) . Hence, (Ik)* = aR = R a and I" a J. This follows,

since Ik c ((I).. Otherwise, I~ = (1"). = aR and then I'J = a R J = aJ C U R = 1'.

Hence, I J c 1. B y Lemma 2.3.10, 1 is a principal right R-ideal ând the contradiction

k = 1 follows- In the case k > 1, the chain of al1 ideds of R is therefore:

To analyze the prime segment of a rank one Dubrovin valuation ring R in a simple

artinian ring Q with finite dimension over its center K , the following result is needed:

Lemma 3.2.2 ( [MMUW], Lemma 7.9 Chapter II) Let R be a Dubrouin

ualuation r ing in a simple artinian r ing Q with Jînite d imens ion ouer its center K .

I f Z ( R ) is f initely generated as a n ideal, then g ( R ) # J(R)* .

The next result shows that in the finite dimensional case the prime segment of a

ranli one Dubrovin valuation ring R is archimedean.

Lemma 3.2.3 If R is a rank o n e Dubrouin ualuation ring i n a simple artinian

ring Q with finite dimension ouer its center I ' , then the prime segment J ( R ) > ( 0 )

m u s t be archimedean.

Proof. By Remark 2.3.7, every prime ideal of R is Goldie. Hence: J ( R) > ( O ) is not

excep tional.

Assume that the prime segment z ( R ) > ( O ) is simple.

Case 1. : J ( R ) i s f ini tely generated as a n ideal

Then, by Lemma 3.2.2, J ( R ) > J ( R ) 2 > ( O ) . But, by Lemma 2.3.10, J ( R ) =

U R = Ra, since Oi(aR) = aRa-' = R. Hence, if g(R)* = ( O ) , then a2R = ( O ) , i-e.,

R = ( O ) . This shows that g ( R ) > ,7(R)2 > ( O ) , a contradiction.

Case 2. : J ( R ) is not f ini tely generated as a n ideal

Let O # x E g(R). Then, RxR # (0) and z ( R ) > RxR > ( O ) , since otherwise

Rx R = J ( R ) is finitely generated as an ideal.

O

To conclude this section we briefiy discuss discrete Dubrovin vaiuation rings.

Definition 3.2.4 A Dubrovin valuation ring R in a simple artinian ring Q is

called discrete i f R is of rank one and J # J 2 .

Let R be a discrete Dubrovin valuation ring of Q. By Theorem 3.2.1, the prime

segment J > ( O ) is archimedean with J # J 2 , D ( R ) - ( J ) % H ( R ) , and n Jn = ( 0 ) .

For, by Lemma 2.3.10, J is a principal as a right R-ideai, Le., J = a R = Ra

for some regular element a E J. The prime segment J > (O) is not simple, since

otherwise J2 = a 2 R = (O), i.e., R = (O). Aiso, J > ( O ) is not exceptional; for

otherwise J = JZ by Lemma 3-12 .

Hence, when Q is a commutative field, then Definition 3.2.4 coincides with the def-

inition of discrete valuation domains in a field, see for example, [Bou'i?], Chapter VI.

Note that a discrete Dubrovin valuation ring R is a non-artinian ring, since otherwise,

by Lemma 1.1.2, every regular element r E R is a unit in R, Le., R = Q. and the

contradiction J = J 2 = ( 0 ) follows.

3.3 Examples

In this section, we discus some examples which show that al1 cases in Theorem 3.1.1

can be realized.

Example 3.3.1 There exists a Dubrouin ualuatéon ring of rank one with an

archimedean prime segment J > ( O ) of the type described i n Theorem 3.21, case

a ) i).

Proof. It follows from the remark after Definition 3.2.4, that any discrete Dubrovin

valuation ring R provides an example to illustrate the case a ) i). In particular, the

ring R = M2 (Z (,)), where p is a prime, is an example of a discrete Dubrovin valuation

ring in the simple artinian ring Q = I&(Q). Also, any discrete rank one commutative

valuation domain R, like Z(,, or the poiver series ring I C [ [ X ] ] over a field K , is an

example for the case a ) i).

O

Example 3.3.2 ( [BS95]) There exists a valuation domain V of r a d one with

quotient field F whose associated valuation has the value group H , a dense subgroup

in (t, +), illustrating the case a) ii) in Theorem 3.2.1.

Proof. Krull's construction from 1932 of a commutative valuation domain with given

commutative ordered group as the value group of the associated valuation, can be

used to illustrate the case a) ii). We take (H, H+) to be any dense subgroup of

(R, +), where H C , is the positive cone of H and as in Example 2.1.1, we construct

a commutative valuation domain V as the localization of the subring I<Hf of the

group ring I i H over a field K at the multiplicatively closed set S = {x ahh E ICH' 1 a0 # O). Le., V = (I<Hf)S-'. The set of al1 principal ideals of V is exactly t h e set

{hl/ 1 h E H ç ) , and hl V = hzV if and only if hl = hz. Then V is a valuation domain

of r a d one in the quotient field F = Q ( K H ) , and the associated valuation on F has

H as its value group. Since H is dense in IR, V is an example that illustrates the case

a) ii) in Theorem 3.2.1.

Example 3.3.3 ( [BS95])

with a s imple prime segment.

There exists a Dubrouin valuation ring R of rank one

Proof. In order to construct a total valuation ring R of rank one with a simple prime

segment? we consider the dense subgroup H = Q of (B, +) and the commutative

valuation domain V = (FH+)ShL constructed in Example 3-32? where F is a field.

We denote by K = Q ( F H ) the field of quotients of the group ring F H . Then

a : + K, defined by a(h) = 2h for h E H, is an automorphism of K compatible

with the valued field ( I i , V). Hence, o induces an automorphism 0 : V -t V with

a(h) = 2h, h E H.

Now; consider the skew-polynomial rings T := V[x, o] c K[x7 a]. Then, K [ x , O] is

a domain with a left and right division algorithm. This implies that K[x, O] is a left

and right PID and hence a noetherian domain. Therefore, K [ x , o] is a left and right

Ore domain. Ço, there exist the classical ring of quotients D := Q(Ii[x, a]).

Coosider the set S = (C l i a i 1 ai E V7 at least one ai is a unit in V). Then, S

is multiplicatively closed and S satisfies the Ore conditions. Therefore, R = TS-' =

{ts-' 1 t E T, s E S} is a subring of D = Q(K[x, O ] ) . Furthermore, R = V [ x , OIS-'

is a rank one total valuation ring in a division ring D. Its nonzero principal right ideals

have the form hl?; h E H+ = O+. Hoivever, xh = xa(k) = $2, i.e., xhR = $ R > hR.

This shows that J ( R ) > (O) is a simple prime segment.

O

Example 3.3.4 ( [Dub93], [BD]) For w e r y nonnegatiüe integer k there en'ts

a rank one total valuation ring whose pr ime segment is exceptional of t ype k.

Proof. The skew field of quotients of the group ring F G of the covering group G over

a skew field F of the group SL(2,R) contains total valuation rings with exceptional

prime segments of type k, for each integer k 2 0. For k = 1, the construction is

described, with more details, in Example 2-3-17. In fact, for al1 k = 0,1,2, ..., the

group G contains certain subgroups Hk with the exceptional positive cone Pk of type

corresponding to k. The cone Pk satisfies the conditions in Theorem 1.3.3 and hence,

there exists a total valuation ring Sk associated with Pk. Now, applying Theorem 1.3.1

to the ring Sk, it follows that Sk is a rank one total valuation ring with the exceptional

prime segment of type k. Note, that for this construction, Dubrovin7s results from

[DubSJ] are needed. No easy construction is known.

0

Sorne results of Chapters 2 and 3 will appear in [BMO].

Chapter 4

Krull Rings

For a commutative Krd l ring, the role that the group of divisorial ideals of a r a d

one Dubrovin valuation ring plays in the previous chapters, is played by the group

of divisors of regular fractional ideals. In this chapter, we discuss commutative and

non-commutative Krull rings. We use the group of divisors of a commutative Krull

ring R with the total quotient ring ri, R # K , to prove an approximation theorem.

A version of this chapter has been published, [Osm99].

4.1 Krullrings

Commutative Prüfer domains play a central role in the study of classical commuta-

tive integral domains. A commutative dornain R is called Pnifer domain if every

non-zero finitely generated ideal of R is invertible. Clearly, commutative PIDs and

commutative Bezout domâins are Prüfer domains. The classical ideal theory of com-

mutative domains has been extended to commutative rings with zero divisors. M.

Griffin [Gri69] introduced the notion of commutative Prüfer rings with zero divisors

and gave many characterizations of that class of rings.

Definition 4.1.1 A commutative ring R is called Prü fer ring i f euery regular

finitely generated ideal of R is inuertible.

Theorem 4.1.2 ( [Gri69], [LM71]) Let R be a commvta t ive ring, and let Ii

be the total quotient ring of R. Then the following s ta tements are equiualenfr

2. For every regular prime ideal P of R, the pair (RLq, [P]R[pl ) is a .Manis d u -

ation pair of AT.

3. For euery regular maximal ideal hf of R; the pair (Rpq, [MI R[,crl) is a Manis

ualuation pair o f K .

4 R is integrally closed and and for al1 a , 6 E R, where a t least o n e of a and b is

regular, ( a , b)" C (an, 6") .

In the non-commutative case, Prüfer rings are defined and studied in [AD901 and

[MMU97].

Definition 4.1.3 ( [AD90]) A prime Goldie ring R is called a right Pmifer ring

every jînitely generated right R-ideal 1 o f R satisfles the equalities

A prime Goldie n'ng R is called a left Prü fer ring i f euery finitely generated lefl

R-ideal J of R satisf ies Lhe equalities

It has been shown in [ADSOI that a prime Goldie ring R is left Prüfer if and only

if it is right Priifer. Also, every Dubrovin valuation ring R in a simple artinian ring

Q is Prüfer ring. For the theory of non-commutative Prüfer rings we refer to [AD901

and [MMUSi'].

The notion of commutative Dedekind domains also has been extended to cornmu-

tative rings with zero divisors.

Definition 4.1.4 A commuta t ive ring R is called Dedekind ring if it is Prüyer

and r-noetherian.

The complete proof of the following characterization can be found in [A090a].

Theorem 4.1.5 ( [AOgOa]) For a commuta t ive ring R the folloz~,ing conditions

are equivalent:

1. Ail regular pr ime ideals o f R are invertible.

2. R is a Dedekind ring.

3. T h e semigroup of regular fractional ideals o f R is a group.

4- Euery regular ideal of R is a product of p r ime ideals.

5. Every regular e lemenl of R is contained on l y in a finite number o f pr ime ideals

of R, and the semigroup of regular ideals can be embedded in a direct product

of ordered cyclic groups

Recall that an R-ideal I of a Dubrovin valuation ring R in a simple artinian ring

Q is called divisorial if I = I', where 1' = n CS, S = O,(I ) and c runs over al1

elements in Q with CS I (Definition 2.3.11). In the case of a rank one Dubrovin

valuation ring R we have the following result:

Proposition 4.1.6 Let R be a rank one Dubrovin valuation ring o f Q and let 1

be a n R-ideal. Then

Proof. Since the only proper overring of R is Q, we have S = O,(1) = R. So,

1- = n CR, where CR 2 1.

Let s E ( R : ( R : I ) [ ) , and c E U(Q) be such that CR 2 1. Then, c-'1 C R, Le.,

C-L E (R : I ) [ . Hence, C-'s E R, Le., s E CR. Therefore, s E 1' and ( R : ( R : l)r), C

1' follows.

Assume that s E 1- and s $ ( R : ( R : I)() , . Then ( R : I ) [ s R, and hence there

exists a regular element c E ( R : I ) [ such that CS 6 R.

Otherwise, for ail regular elements c E ( R : 1)[, we have cs E R- But, by

Lemma 1.1.8, ( R : I )[ is a left R-ideal. Then, by Definition 1-1.7, there exists

c E U(Q) with ( R : I)/c C R. Since ( R : I ) [ n U(Q) # 0 (also by Definition 1.1.7),

(R : I ) / c contains a regular element of R and is therefore an essential left ideal of

R. Thus: by Theorem 1.1.5, (R : I )[c is generated by a set {c i } of regular elements;

so, as a left R-submodule of Q, (R : 1); is generated by the set {tic-') of regular

elements. So, ( R : 1 ) [ ~ C R, a contradiction.

Then, s 6 C-'R. On the other hand, c E (R : I ) [ implies CI C R and 1 c-IR.

Since s E 1': this implies s E c-'R.

O

Remark 4.1.7 Proposition 4.1.6 holds in the more general situation when R is

an order i n a simple artinian ,ring Q and for an R-ideal 1, the +-operation i s dejned

by I* = n CR, where CR > 1.

In the commutative case, an analogous operation is defined in the following way.

Let R be a commutative ring; we denote by K the total quotient ring of R and by

F ( R ) the set of al1 regular fractional ideals of R. If A, B E F ( R ) , then ( A : B) =

{x E I< 1 XB C A) is also in F ( R ) . On the set F ( R) we define an equivalence relation

- by: A - B w ( R : A) = (R : B) . The set of al1 equivalence ciasses is denoted by

D(R), and the class containing A E F(R) is denoted by div(.4), the diuisor of A. If

A E F(R) we define A to be the set (R : (R : A ) ) E F(R) . A regular fractional ideal

A of R is diuisorial if A = A.

Remark 4.1.8 If R is a commutative integral dornain, then A is equal to the inter-

section of al1 nonzero principal fractional ideals containing A. This is no longer true

if R is a commutative ring with zero divisors, see [AM85].

However, ive still have a satisfactory theory of divisors which is sirnilar to the the-

ory of divisors for integral domains. For A, B E F(R) we define div(A) + div(B) =

div(AB). Under this operation, D ( R ) is a commutative semigroup with identity ele-

ment div( R). We Yeso define a relation 5 on D( R) by setting div(A) 5 div( B) if and only if B C - A. Then D ( R ) is a partially ordered semigroup. D(R) is a group if and only if R is

completely integrally closed, see [HucSS], Th.S.4.

R-Kennedy [Ken731 introduced the class of commutative Krull rings with zero

divisors and developed t heir t heory of divisors.

Definition 4.1.9 ( [Ken73]) Let R be a commutatiue ring with the total quotient

ring K such that R # ïï. Then R is called a I<rull ring if there exists a Jamily

{(V,, Pa) 1 a E I } of discrete rank one unluation pairs of K with associated unluations

{va 1 o E I } such that

(K2) v,(a) = O almost everywhere on I for each regular element a E Iir and each

Pa is a regular ideal of Va.

The following theorem provides the main characterization for commutative Kru11

rings with zero divisors.

Theorem 4.1.10 ( [Ken73], Conjec. 3.26 and [Mat82], Th 5 . ) Let R be

a commutative ring which is not equal to its total quotient ring. Then R is a Kmll

ring i f and only i f R is comp le t e i y integrally closed and each nonemp ty collection of

d iv isonal ideals of R h a s a m a x i m a l e k m e n t -

Note that if R is an integral dlomain different from its quotient field, then R is a Iirull

domain if and only if R is a IKrull ring.

Corollary 4.1.11 Everp commutative Dedekind ring is a I\'rull ring.

Pro of. If R is a Dedekind rin:g, then R is r-noetherian and Prüfer. By Theorem 4-12,

R is integrally closed, and h~ence, by Proposition 1.1.14, R is completely integrally

closed. So, by Theorem 4.1.1 0, R is a KruU ring.

O

We mention now some reszults which show that commutative Kru11 rings with zero

divisors not only share many .cornmon properties with Kru11 integral domains but also

show that there are differences. The following result is an immediate consequence of

Proposition 1.1.14 and Theorem 4.1.10.

Proposition 4.1.12 ( [AOC30b], Prop.2.3.) Le t R be a n r-noetherian ring that

k distinct fr0.m i t s total quodient ring. Then R is- a Kru11 ring and o n l y if R is

integrally closed.

Proposition 4.1.13 ( [Bou'72], Propl3, p. 488) Let R be a IiFrull domain

and XI, 2 2 , . . . , Zn be i n d e t e m i n a t e s . Then the r ing R [ x i , rz, . . . , x,] is a l h l f Rng.

Note, that in the case of rings with zero divisors this is not true. For example, the

ring R = Z @ Z/Zpn , where g is a prime and n 2 2 , is a Kru11 ring, but R[X] is not

a Kru11 ring since it is not completely integrally closed.

Recall that a commutative ring R is called a Marot r ing or a ring with the property

(P) if every regular ideal is geonerated by regular elements.

Exarnple 4.1.14 A Kmll ring R which is not Marot.

Proof. Let D = Z [ F 5 ] . Then D is a Dedekind domain with a maximal ideal

144 = (1 + J=j)Z + 2 2 which is not principal but hl2 = 2D is principal. Let A be

the D-module @{D/Q 1 Q a maximal ideal of D , Q # M } , and let R be the ring

D(+)A , the idealization of the D-module -4, Le., (d, a ) + (8, a') = (d + dl, a + a') and

(dl a)(dr, a') = (dd', ad' + a'd) for all a , a' E -4, d, d' E D. For more details see, for

example, [HucSS]. Shen:

1. The ideal P = M ( + ) A of R is not principal.

2. P is the unique regular prime ideal of R.

3. The set of a11 regular ideals of R is the set of the form (Pn}N, .

4. The ideal P is invertible.

5 . The set of al1 regularly generated ideals of R is {tn R):?, , for some t E R.

Hence, by Theorem 4.1.5 aad Corollary 4.1.1 1, R is a Kru11 ring. But. the ideal P is

a regular ideal which is not generated by regular elements since P is not a principal

ideal and the set of al1 regularly generated ideals of R contains only principal ideals.

Hence, R is not a Marot ring.

Finally, at the end of this section we briefly discuss how discrete Dubrovin valu-

ation rings are used in studying non-commutative Iirull rings.

Non-commutative Kru11 rings have been defined and studied by H. Marubayashi,

[Marï5], [Mar76], [Maris], M. Chamarie, [ChaSl] and N. Dubrovin, [DubSlb]. In

[DubSlb], Dubrovin uses discrete Dubrovin valuation rings in a simple artinian ring

to define non-commutative Kru11 rings.

Definition 4.1.15 ( [Dubglb]) A s~br ing R in a simple artinian ring Q is

called a non-commutative I h l l rîng if there exists a family {R; 1 i E A} of discrete

Dubrovin ualuation rings of Q such that

( K 2 ) for each regular element q E Q , the equation qR, = R, holds for almost al1 i in

A;

(K3) fo r any finite set {io, il, - - ,in) in A, there exits an element q E Q such that

q r 1 (rnod ,?(R;,)), q O (mod g(&,)) for 1 < t 5 n, and q E Ri for al1

other j E A.

It follows from the results in [Dubglb] that the class of non-commutative Krull

rings defined in Definition 4.1.15 coincides wit h the class of non-commutative Krull

rings defined in [Mar75], where valuation rings are not used.

4.2 Essential valuations of Krull rings

We denote by R a commutative Kru11 ring with total quotient ring K. Then D ( R )

is a group. Let M(R) be the set of al1 maxima1 divisorial integral ideals of R. If

P E li1 (R) then div(P) is a minimal positive element of the group D(R). We denote

the set {div(P) P E M ( R ) } by P(R) . Each divisor div(.4) E D ( R ) can be written

uniquely as:

If div(.4) = C np!J3 and div(B) = C n$$3, then div(r2) 5 div(B) if and only if

n p 5 rnp for a11 13 E P(R) .

For each divisor E P ( R ) we define a map from F(R) to the set of integers Z

by v p ( A ) = n.p, where div(A) = C nu U. The map vip has the following properties:

a) vg(-4B) = vy(A) +vv(B) for all A, B E F(R) ;

b) u9(-4 + B ) = min{vp(.4),vp(B)) for al1 A, B E F ( R ) ;

c ) A C B is equivalent to vT(A) 2 vp(B) for ali !J3 in P ( R ) , where A, B E F(R) -

and B = B.

The family of the valuations {va : !$l E P ( R ) ) defined by:

( v x E K ) vp(x) = sup { v p ( ~ x + A) 1 A E F(R))

is said to be the family of essential valuations of a Kru11 ring R. The ring of the

valuation v.p is denoted by Vp and the positive ideal by PT. The essential valuations

of a Kru11 ring R are discrete of rank one and they d e h e the ring R, Le., R = n{VP 1 <P E P(R)} and q ( a ) = O for almost al1 !+3 in P(R), for al1 a E C K ( 0 ) , see [MattZ].

Essential valuations of a Krull ring have the following properties:

Proposition 4.2.1 ( [AOgOb], Prop.2.5) If x E C K ( 0 ) a n d 73 E P ( R ) , t h e n

v ~ ( x ) = v ~ ( R x ) .

Proposition 4.2.2 ( [AOSOb], Prop.2.6) Let A b e a divisorial ideal o f R- and

div(i2) = z{np<P 1 E P ( R ) ) . T h e n a E A i f and only ifvp(a) 2 np.

Proposition 4.2.3 ( [AOgOb], Prop.2.7) Let np E % wi th almost all n.p = O

for 73 E P ( R ) , and set A = {x E I< ( vp(x) > np ( V q E P ( R ) ) ) . T h e n A is a

regular divisorial ideal, and div(A) = C{nP'iJ3 1 Q E P ( R ) ) .

Proposition 4.2.4 ( [AOgOb], Prop.2.8) Let = div(P) E P(R), where P

is a max imal diuison'al ideal of R. Then P is a m i n i m a l regular pr ime (r-pr ime)

ideal of B. T h e ualuation r ing of vp 2s equal to R[pl, and the positive ideal of vp is

[Pl R[P]

Coroilary 4.2.5 ( [AOgOb], Cor.2.9) The family {P 1 div(P) E P(R)} o f al1

max imal divisorial ideals o f R is precisely the family of al2 m i n i m a l regular pr ime

ide& o f Rr

4.3 An approximation theorem for Kru11 rings

Approximation theorems are very cornmon in many areas of rnathematics, for exam-

ple, in the valuation theory of fields and commutative rings, in multiplicative ideal

theory and in the theory of partially ordered abelian groups. They are used as a

tool in studying topological structures and play an important role in the description

of the division class group. The Chinese Remainder Theorem is the oldest form of

an approximation theorem for congruences. In this section we prove an approxima-

tion theorem for essential vduations of commutative Kru11 rings mith zero divisors.

Throughout, rings are assumed to be commutative with identity 1.

Theorem 4.3.1 Let R be a commuta t ive I h l l ring wi th to tal quotient ring Ii

such tha t R # I( and {vP 1 73 E P ( R ) } is the family of essential valvations of

R. { u ~ , , v.p,, - - . , vv,) is a finite set of essential valuations of the ring R a n d

(ml,m2, . . . ,m,) E Zn7 then there exists an elernent z E Ii s v c h that vv,(z) = mi

for aM i E {1,2,. . . , n) and up(x) 2 O /or al1 other E P(R).

Proof. It suffices to prove the theorem in the case where at most one of the mi is

not equal to O.

Let m2 = m3 = . . . = rn, = O . If mi = O then x = 1.

Case 1. Let rn > O and pi = div Pi, i = 1,2, . . . , n be minimal positive elements

of the group D(R), where the Pi are maximal divisorial integral ideals of R. By

Proposition 4.2.4, Pi is a minimal regular prime ideal of R? i E {1,2,. . . , n}. Then

P r 1 and PY'+' are divisorial integral ideals of R. Also:

Othenvise, by [Bou7'2, Proposition 2, Ch.11, SI], PrL C P;~I+' or PTL C Pi, for some

i E {2,3 , . . . , n). But then div(fi) 5 O or Pl = Pi, which is impossible.

P,"' . Hence, by the

Assume that vp, (x) 2 ml + 1. Then, there exists an element L E F ( R ) such that

v.pl (Rx + L ) 1 1 + 1 - We can assume that L C R. For, there exists a regular element

d E R such that dL C R. Shen v P ( R r + L) > - V~(P;LL+') for al1 !)? E P(R), Since

Rx + L C R implies uw(Rx f L) 3 O = V ~ ( P ~ ~ + ~ ) for al1 !)? # pl. By the property

c) of the rnap vp, we have Rx + L C PT'+^, i.e., x E P r L + ' , a contradiction.

Therefore, wp,(x) < ml + 1, Le., vp,(s) = ml.

It remains to prove that vpi (x) = O = mi for al1 i E {2,3, . . . n}. Since x E

p;"' C R = n l$ L L$ for al1 (p E P ( R ) , vp(s) 2 O for all 73 E P(R) . t\ssurne that

vp,(x) > O for some i E (2 ; 3 ? . . . , n}. Then x belongs to [Pi] RIPiI, by Proposition 4.2.4.

Hence, x E Pi, which is a contradiction.

Case 2. Let m l < O and let r E Reg(PFm'). By Proposition 4.2.1, v.p,(r) =

vp,(Rr) 2 -mi > O. Then vp,(r-') < O. We prove that the set {<P E P ( R ) 1 vp(r-l) < O} is finite. If not, then the divisor of Rr-' E F ( R ) in its decomposition

div(Rr-') = C (vp(~r- ' ) ) !J3 has infinitely rnany coefficients different from O. WEP(R)

This is impossible since D(R) is a free Z-module with a free bases P(R) .

Let {QI, %?,. . . . %,} be the set of all elements !J3 in P ( R ) such that vP(r-') < 0.

Then vP(r-') = O for al1 E P ( R ) \ {P l , va,, . . . ,Pa,}, since r E Pym' c R implies

vP(r-') 5 O for ail <p E P(R) .

Consider the family { v ~ , , vva,, . - - , vqat) of essential valuations of the ring R and

(upi ( r ) f mi, vp,, ( r ) , . . - , vp., ( r ) ) E Z'. Since vP1 ( r ) + mi > O and upai ( r ) > O for

al1 i E (2,. . . , t ) we can apply Case 1. Hence, there exists an element xi E R such

chat UV, ( x i ) = ugl ( r ) + ml7 u-poj(xl) = vVmi ( r ) for al l i E (2,. . . , t ) and vP(x i ) 2 O

for ail other <p E P ( R ) . Shen x = r-L 11 E K is an element satisfying the required

condition.

O

The element x constructed in the proof of the Theorem 4.3.1 is not necessarily

regular. For the class of additively regular rings it follows from the next result that

the elernent x can be chosen to be regular. This result is also an easy consequence

of Theorem 4.3.1. But first, we recall the definition of an additively regular ring. A

ring R is additively regular if for each c in the total quotient ring A- of R, there exists

u E R such that z + u f &(O).

Theorem 4.3.2 Let R be a n additively regular Km11 ring such that R # K, and

let {vP : !J3 E P ( R ) ) be the Pmi ly of essential valuations of R. If {uv17 v y 2 , . . . , vpn)

is a finite set of essential valuations and if(rni, m 2 , . . . ,m,) E Zn, then there exists a

regular element x E IlF such that vpt(x) = mi for al1 i E {1,2,. . . n) and vp(x) 3 O

for al1 other E P ( R ) .

Proof. In the proof of Theorem 1.3.1, Case 1, if x E PLL \ (p;LLC1 U P2 U P3 U - - U P,),

then, by the additive regularity of R, for b E Reg (P;"~" n P2 P3 n - . - n P,),

there exists an element u E R such that t = x + bu is regular and t E Pr1 \ ( P;"LC' (J PZ U - U P,). The element t satisfies the required conditions.

O

This allows us to characterize divisorial fractional ideals of an additively regular

Krull ring in terms of principal fractional ideals in the same way as in the case of

domains, see [Bou72], Ch. VII, 55, Corollary 2.

Corollary 4.3.3 Let R be an additiuely regular K m l l ring and let -4: B,C be

divisorial fractional ideals of R such that -4 E BI Then there exists a n e lement x E 1ir

such that

Proof. Let {vv 1 !J3 E P(R)) be the family of essential valuations of a Kru11 ring R

and let

Then A C B implies div(A) 2 div(B) arid hence rny 2 n y for al1 73 E P(R). Let

Ji = {<p E P ( R ) ( r n ~ > np}. Also, there exists a finite subset Jz of P ( R ) such

that pp = m p = O for aii <P E P ( R ) \ J2. Let J = J I U J2. Then {vg [!J3 E J ) is a

finite family of essential valuations of the ring R and {ptP - rnp 1 !J3 E J ) is a finite

family of integers. By Theorem 4-32? there exists a regular element x E K' such that

vp(x-') = p p - mg for al1 'J3 E J and ve(x-l) > - O = pp - rnp for al1 !JJ E P(R) \ J.

Hence:

since vp(x) = vv(Rx).

Suppose that a E A. By Proposition 4.2.2, vp(a) 2 m p =

S U P { ~ ~ : V ~ ( R ~ ) + p p ) for al1 E P(R) . Therefore, vg(a) 2 n.p and vy(a) > - ?p(Rx) + p p for al1 73 E P(R) . On the other hand, Rx E F(R) and

Furt hermore, since the fractional ideal R x is invertible, Rx- C is a divisorial ideal of the

ring R. Applying Proposition 4.2% again, we get a E B fi RxC, i.e., A C B n RxC.

Conversely, for a E B n RxC, v y ( a ) 2 nq and v P ( a ) 2 vq(Rz) + pp for al1

E P ( R ) - Hence, vp (a ) 2 sup InT, vg(Rx) + pp) = m p for all !J3 E P ( R ) Le., a E A

by t h e s a m e argument.

O

Corollary 4.3.4 Let R be an additiuely regular fi-mil Rng such that R # I i and

let A be a fractional regular ideal of R. Then A i s a diuisorial ideal of R i f and only

$il is the intersection of two fractional principal ideals of R.

- Proof. I f A i s a divisorial ideal, then A = A C n { R a 1 a E &-(O). A C R a ) . Let

B = R a , a E CA-(O) be any regular principal fractional ideal of R such that A C B

and let C = Rb, b E CA-(O). Then, by Corollary 4.3.3, t here exists a regular element

x E Ii S U C ~ that A = B n RXC = R a n Rxb.

Conversely, if -4 = Ra Rb where a, b E CK (O), then Ra n Rb = A C 2 5 -

n{Ry 1 y E CA-(O), A C Ry) C R a n R b = A, i.e., A= A.

O

Bibliography

J.H. Alajbegovic and N.1 Dubrovin, Nonco~mmutati~ve Prüfer rings? J . Al-

gebra 135 (1990), 165-176.

D.D. Anderson and R. Markanda, Unique factorkation mngs with zero

divisors, Houston J . Math. 11 (1985), 423426.

J.H. Alaj begovic and J . MoCkof, Approximation Theorerns in Com.mutative

Algebra,Classical and Catego,rical Methods, Kluwer -4cademic Publishers,

Dordrecht/Boston/London, 1992.

J.H. Alajbegovic and E. Osmanagic, Dedekind and Km11 rings with zero

divisors, Znanstv. Rev. 1 (1990), 7-20.

J.H. Alaj begovic and E. Osmanagic, Essential valuations of Km11 mngs

with zero diuisors, Comm. Algebra 18 (1990), 2007-2020.

H.H. Brungs and N.I. Dubrovin, Classification of chain rings, preprint.

R.A. Beauregard, Overrings oJ Bezout dornains, Canad. Math. BuII. 16

( l g ï 3 ) , 4'754'77.

H . H . Brungs and J . Grater, Extensions of raluation rings in centml simple

algebras, Trans. Amer. Math. Soc. 317 (1990), 287-302.

[BMO]

[Bou721

[BruSG]

[BS95]

[B TT61

[BTSS]

[Chas11

[Coh63]

[Coh85]

[DD96]

[DubM]

H.H. Brungs, H. Marubayashi, and E. Osmanagic, A classifcation of prime

segments in simple artinian rings, Proc. Amer. Math. Soc., to appear.

N. Bourbaki, Elements of Mathematics: Commutative Algebra (Ch. 6),

Herman and Addison-Wesley, Paris-Reading, 1972.

H.H. Brungs, Bezout domains and rings with a distributive lattice of right

ideals, Canadian J. Math. 38 (1986): 286-303.

H.H. Brungs and M. Schroder, Prime segments of skew fields, Canadian J.

Math. 47 (1995), 1148-1176.

H.H. Brungs and G. Torner, Chain rings and prime ideals, Arch. Math.

27 (1916), 253-260.

H.H. Brungs and G. Torner, Ideal theory of right cones and associated

.rings, J. Algebra 210 (1998), 145-164.

M. Charnarie, Aroncornrnutative Kru11 rings (French), J. Algebra 72 (19811,

210-222.

P.M. Cohn, iVoncommutative unique facton-zation domains, Trans. Amer.

Math. Soc. 109 (1963), 313-331.

P.M. Cohn, Free Rings and their Relations, Acadernic Press, NewYorli,

1985.

T.V. Dubrovina and N.I. Dubrovin, Cones in gmups, Math. Sbornik 187

(1996), 59-74,

N.I. Dubrovin, Noncommutative unluation rings? Trans. Moskow Math.

SOC. 45 (19S4), 2'73-2S7.

[DubS5] N.I. Dubrovin, Noncommutative valuation rings in simple Jinite-

dimensional algebras ouer a field, Math. USSR S bornik 5 1 (l985), 493-50.5.

[Dubgla] N.I. Dubrovin, Noncommutatiue arithmetic rings, Fh-D. thesis, Pol. Insti-

tute? Vladimir, 1991, Thesis.

[DubSlb] N.I. Dubrovin, Noncommutative l''ru11 rings (Russian), Sovieth. Math. (Iz.

VUZ) 35 (1991), 11-16.

[Dub93] N.I. Dubrovin, The rational closure of group rings of l ep orderable groups,

Math. Sbornik 187 (1993): 3-48.

[Fuc63] L. Fuchs, Partially Ordered Alge6raic Systems, Pergamon Press, New York,

1963.

[Gi172] R. Gilmer, iI1ultiplicatiue ldeal Theory, Marcel Dekker, New York, 1972.

[GraEa] J. Grater, The Defektsatr for central simple algebras, Trans. Amer. Math.

SOC. 330 (1992), 823-843.

[Gra92b] J. Grater, Prime PI-rings in which finitely generated right ideals are prin-

cipal, Forum Math. 4 (19921, 447-463.

[Gri69] M. Griffin, Prufer rings with zero diuisors, J. Reine Angew. Math.

239/240 (1969), 55-67.

[HerGS] 1. Herstein, ~Voncommutatiue Rings, The Carus Mathematical Mono-

graphs, No. 15, The Mat hematical Association of America, 196s.

[HucSS] J .A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker Inc.,

New k-ork and Basel, 198s.

[Ken731 R. Kennedy, .4 generafization oflCmfl domains to rings with zero divisors,

Ph.D. thesis, Univ. Missouri-Kansas, 1973, Thesis.

[KopSS] V.M. Kopytov, Right Ordered Groups, J. Wiley, 1998.

[Km321 W. Knill, Allgen~eine Bewertungstheon'e, J. Reine Angew Math. 167

(1932), 160-196.

[LM711 M.D. Larsen and P.J. McCarthy, Multiplicatiue Theory of Ideals, Academic

Press, New York and London7 1971.

[Man671 M.E. Manis, Extension of valuation theory, Bull. Amer. Math. Soc. 73

(1967), 735-756.

[Mas751 H. Marubayashi, Noncommutative Krull rings, Osaka J. Math. 12 (1975),

703-714.

[Mar'iG] H. Marubayashi, On bounded I h l l prime rings, Osaka J. Math. 13 (1976),

491-501.

[Maris] H. Marubayashi, A characterization of bounded Km11 prime rings, Osaka

J. Math. 15 (197S), 13-20.

[hIatSl] R. Matsuda, Iironeckerfunction rings, Bul. Fac. Sci. Ibaraki Univ. Ser. A,

No. 13 (19S1), 13-24.

[MatSZ] R. Matsuda, On Kennedy's problems, Comment. Math. Univ. Sancti. Pauli

31 (19S2), 143-145.

[Mat%] R. Matsuda, Generalization of multiplicative ideal theory to commutative

rirzgs .with zero diuis'o,rs, Bull. Fac. Sci. Ibaraki Univ. Ser. A, No. 17 (1985),

49-101.

[MMU97] H. Marubayashi, A. Miyamoto, and A. Ueda, Non-commutative Valuation

Rings and Semiheridetary Orders, Kluwer Acad. Publ., Dordrecht, 1997.

[Mor89] P.J. Morandi, Valued functions on central simple algebras, Trans. Amer.

Math. Soc. 315 (19S9), 605-622.

[MW891 P.J. Morandi and A.R. Wadswort h, Integral Dubrovin valuation h g s ,

Trans. Amer. Math. Soc. 3115 (19S9), 623-640.

[Osrn99] E. Osmanagic, On a n approximation theorem for I h l l rings with zero

divisors, Comm. Algebra 27 (1999), 3647-365'7.

[PiesSI R.S. Pierce, Associatiue Algebras, Springer-Verlag, New York-Heidelberg-

Berlin, 1983.

[RMS Cl J .C. Robson and J-C. ;\/lcConnell, AJoncommutatiue Noetherian Rings,

AWiley-Interscience Publication, 1987.

[Rob67a] J-C. Robson, Artinian quotient rings, Proc. London Math. Soc. 17 (196'i),

600-616.

[Rob67b] J-C. Robson, Rings in which finitely generated ideals are principal: Proc.

London Math. Soc. 17 (1967), 617-63s.

(Sch451 O.F.G. Schilling, Noncom~mutatiue valuations, Bull. Amer. Math. Soc. 51

(1945), 297-304.

[Sch50] O.F.G. Schilling, The Theory of Valuations, Amer. Math. Soc., 1950.

[WadSS] A.R. Wadsworth, Dubrovin vahation rings and Henseliration, Math. -4nn.

283 (19S9), 301-32s.

Index

extension

algebra

central simple, 16

division, 20

annihilator, 6

approximation t heorem, 73

common denominator, 6

cone, 17

divisor, 67

dornain

noet herian, 62

Bezout, 22, 24

left Ore, 62

left PID, 62

Ore, 20

PID, 12

Prufer, 23

efement

aimost integral, 11

integral, 11

of vduation ring, 16

total invariant, 21

Goldie

dimension, S, 9

ring, S

Theorem, S

€PUP

linearly ordered, 17

right ordered, 17

universal covering, 41

Holders Theorem, 49

ideal

R-ideal, 9

completely prime, 18, 40

divisorial, 36, 68, 72

essential, S

fractional, 67

Goldie prime, 32, 40, 46

idempotent, 43

integral R-ideal, 9

P-ideal, 18

positive, 13

prime, 1s regular, 12, 69

jump, 52

localization, 23, 3 1

module

essential, S

uniform, S

order, 6

Bezout, 31, 33

local Bezout, LS

Ore

condition, 5 , 63

regular set, 6

set, 5 , 35

prime segment, 45, 51

archimedean, 53, 53

exceptional, 53, 54

simple, 53, 54

ring

additively regular, 12, 75

ari t hrnetic, 1

Bezout, 23

chain, 10

commutative ICrull, 68, 70

completely integraily closed, 11, 12,

6s

Dedekind, 66, 69

Dubrovin valuation, 16, 20, 25, 26,

52

Dubrovin valuation discrete, 60, 61

Dubrovin valuation of r a d one, 56

integrally closed, 12, 69

invariant, 15

large quotient, 11

Local, 22

Marot, 12, 69, 70

n-chain, 10, LS

noetherian, 12

noncommutative arithmetic, 17

of quaternions, 70

PrÜfer, 65

prime Goldie, 31, 31

quotient, 6

r-noetherian, 11

semiprime Goldie, 24

simple, 5

simple artinian, 28, 31, 51

total quotient, 67

total valuation, 15, 10, 31

valuation, 13

with few zero divisors. 12

skew field

associated with cone, 18

subgroup

convex, 52

valuation

discrete, 13

essential, 72

Krull, 13

Manis, 14

of rank one, 13

on a division ring, 15

Schilling's, 15

trivial, 13

valuation pair, 14, 65

value group, 13

Wedderburn's Theorem, 5