Commutative Algebra - Bourbaki

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N I C O L A S BOURBAKI

Elements of Mathematics

CommutativeAlgebra

Originally published asELEMENTS DE MATHEMATIQUE, ALGEBRE COMMUTATIVE

1964, 1965, 1968, 1969 by Hermann, Paris

ISBN 0-201-00644-8

Library of Congress catalog card number 78262

American Mathematical Society (MOS) SubjectClassification Scheme (1970) : 13-02,

Printed in Great Britain

1972 by Hermann. All rights reservedThis book, or parts thereof, may not be reproduced in any form without

the publisher’s written permission

TO TH E READER

1. This series of volumes, a list of which is given on pages ix and x, takes upmathematics at the beginning, and gives complete proofs. In principle, itrequires no particular knowledge of mathematics on the reader’s part, but onlya certain familiarity with mathematical reasoning and a certain capacity forabstract thought. Nevertheless, it is directed especially to those who have agood knowledge of at least the content of the first year or two of a universitymathematics course.

2. The method of exposition we have chosen is axiomatic and abstract, andnormally proceeds from the general to the particular. This choice has beendictated by the main purpose of the treatise, which is to provide a solidfoundation for the whole body of modern mathematics. For this it is indis -pensable to become familiar with a rather large number of very general ideasand principles. Moreover, the demands of proof impose a rigorously fixed orderon the subject matter. It follows that the utility of certain considerations willnot be immediately apparent to the reader unless he has already a fairlyextended knowledge of mathematics; otherwise he must have the patience tosuspend judgment until the occasion arises.

In order to mitigate this disadvantage we have frequently inserted examplesthe text which refer to facts the reader may already know but which have

not yet been discussed in the series. Such examples are always placed betweentwo asterisks: * . . . Most readers will undoubtedly find tha t these examples

help them to understand the text, and will prefer not to leave them out,even at a first reading. The ir omission would of course have no disadvantage,from a purely logical point of view.

This series is divided into volumes (here called The first six are numbered and, in general, every statement in the text assumes as

known those results which have already been discussed in the preceding

TO THE READER TO THE READER

to completeness ; in particular, references which serve only to determinequestions of priority are almost always omitted.

As to the exercises, we have not thought it worthwhile in general to indicatetheir origins, since they have been taken from many different sources (originalpapers, textbooks, collections of exercises).

References to a part of this series are given as follows:

If reference is made to theorems, axioms, or definitions presented in the samesection, they are quoted by their number.

If they occur in another sectionof the same chapter,this section is also quotedin the reference.c) If they occur in another chapter in the same Book, the chapter and section arequoted.d) If they occur in another Book, this Book is first quoted by its title.

signifies “Summary Results the Theoryof Sets.”The Summariesof Results are quoted by the letter R: thus Set Theory,R

volumes. This rule holds within each Book, but for convenience expo-sition these Books are no longer arranged in a consecutive order. At the begin -ning of each of these (or of these chapters), the reader will find a preciseindication of its logical relationship to the other Books and he will thus be

to satisfy himself of the absence of any vicious circle.

The logical framework of each chapter consists of the definitions,the axioms,the of the chapter. These are the parts that have mainly to be

borne in mind for subsequent use. Less important results and those which canbe deduced from the theorems are labelled as

etc. Those which may be omitted at a first readingare printed in small type. A commentary on a particularly important theoremappears occasionally under th e name of

To avoid tedious repetitions it is sometimes convenient to introduce nota -tions or abbreviations which are in force only within a certain chapter or acertain section of a chapter (for example, in a chapter which is concerned onlywith commutative rings, the word would always signify “commutative

Such conventions are always explicitly mentioned, generally at thebeginning of the chapter in which they occur.

6. Some passages in the text are designed to forewarn the reader againstserious errors. These passages are signposted in the margin with the sign

(“dangerous

7. The Exercises are designed both to enable the reader to satisfy himself tha the has digested the text and to bring to his notice results which have no placein the text but which are nonetheless of interest. The most difficult exercisesbear the sign

8. In general, we have adhered to the commonly accepted terminology,except where there appeared to be good reasons for deviating from it.

9. We have made a particular effort always to use rigorously correct language,sacrificing simplicity. As far as possible we have drawn attention in th e

to abusesof language,without which any mathematical text runs the risk of pedantry, not to say unreadability.

Since in principle th e text consists of the dogmatic exposition of a theory,contains in general no references to the literature. Bibliographical references

are gathered together in Historical Nates,usually at the end of each chapter.These notes also contain indications, where appropriate, of the unsolvedproblems of the theory.

The bibliography which follows each historical note contains in generalonly those books and original memoirs which have been of the greatest impor -tance in the evolution of the theory under discussion. I t makes no sort of

CONTENTS

OFTHE ELEMENTS OF MATHEMATICS SERIES

I. T HEORY OF SETS

1. Description of formal mathematics.cardinals; natural numbers. 4. Structures.

2. Theory of sets. 3. Ordered sets;

11. ALGEBRA

1. Algebraic structures. 2. Linear algebra. 3. Tensor algebras, exterioralgebras, symmetric algebras. 4. Polynomials and rational fractions.5. Fields. 7. Modules over principal idealrings. 8. Semi -simple modules and rings. 9. and quadraticforms.

6. Ordered groups and fields.

111. GENERAL TOPOLOGY

1. Topological structures. 2. Uniform structures. 3. Topological groups.4. Real numbers. 5 . One -parameter groups. 6 . Real number spaces,

and projective spaces. 8.numbers 9 Use of real numbers in general topology 10 Function

7. The additive groups R " .

CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES

LIE GROUPS AND LIE ALGEBRAS

. Lie algebras .Tits systems .

2. Free Lie algebras . 3. Lie groups .5. Groups generated by reflections .

4. Coxeter groups6. Root systems .

COMMUTATIVE ALGEBRA

. Flat modules .logies .. .

2. Localization . 3. Graduations, filtrations. and topo -. Integers .4 . Associated prime ideals and primary decomposition .

7. Divisors .SPECTRAL THEORIES

1. Normed algebras . 2. Locally compact groups .D IFFERENTIABLE AND ANALYTIC MANIFOLDS

Summary of results .

CONTENTS

...............................................To THE READEROF THE ELEMENTS OF MATHEMATICS SERIES . . . . . . . . . . . . . .

INTRODUCTION ................................................

I. FLAT MODULES ................................... Diagrams and exact sequences ........................... Diagrams .......................................... Commutative diagrams .............................. Exact sequences ...................................

4 The snake diagram. ................................ Flat modules .........................................

. ..........................1 Revision of tensor productsM-flat modules

3. Flat modules ......................................4. Examples of flat modules5. Flatness of quotient modules

6. Intersection properties ..............................7. Tensor products of flat modules ......................8. Finitely presented modules ..........................9. Extension of scalars in homomorphism modules . . . . . . . .

10. Extension of scalars: case of commutative rings . . . . . . . .. Interpretation of flatness in terms of relations . . . . . . . . . .

. Definition of flat modules . . . . . . . . . . . . . . . . . . .2. Tensor products of faithfully flat modules .............

. ...............................................................

........................

3. Faithfully flat modules .................................

. Change of ring ....................................

111234

10121415171920222225

27273031

CONTENTS CONTENTS

4. Restriction of scalars ............................... 315. Faithfully flat rings ................................ 326. Faithfully flat rings and finiteness conditions . . . . . . . . . . . 347. Linear equations over a faithfully flat ring . . . . . . . . . . . . 35

. Flat modules and “Tor” functors ........................for 1 ...........................................

Exercises for ...........................................Exercises for ...........................................Exercises for ........................................... 50

. LOCALIZATION .................................... 51. Prime ideals .......................................... 51

51. Relatively prime ideals ............................. 53

. Rings and modules of fractions .......................... 551. Definition of rings of fractions ....................... 552. Modules of fractions ............................... 603. Change of multiplicative subset ...................... 644. Properties of modules of fractions .................... 675. Ideals in a ring of fractions .......................... 706 . Nilradical and minimal prime ideals . . . . . . . . . . . . . . . . . . 737. Modules of fractions of tensor products and homomorphism

modules ........................................ 758. Application to algebras ............................. 779. Modules of fractions of graded modules ................ 78

3. Local rings . Passage from the local to the global . . . . . . . . . . . 801. Local rings ....................................... 802. Modules over a local ring ........................... 823. Passage from the local to the global . . . . . . . . . . . . . . . . . . 87. Localization of flatness ............................. 91

5. Semi -local rings ................................... 92

949497

1. Definition of prime ideals ............................

. Spectra of rings and supports of modules ..................1. Irreducible spaces ................................... topological spaces ......................... prime spectrum of a nng ....................... 98

4. support of a module ............................

3 5. Finitely generated projective modules . Invertible fractionalideals ............................................... Localization with respect to an element ...............

2. Local characterization of finitely generated projectivemodules ........................................

3. Ranks of project ive modules .........................4. Projective modules of rank 15. Non -degenerate submodules .........................6 . Invertible submodules ..............................7. The group of classes of invertible modules . . . . . . . . . . . . .

........................

108108

109111114116117119

1 ........................................... 121

Exercises for 2 ........................................... 123

Exercises for 3 ........................................... 136

Exercises for 4 ........................................... 140

Exercises for 5 ........................................... 146

CHAPTER . GRADUATIONS . F ILTRATIONS AND . . . . . . . . . 1551. Finitely generated graded algebras . . . . . . . . . . . . . . . . . . . . . . . 155

1. Systems of generators of a commutative algebra . . . . . . . .2. Criteria of finiteness for graded rings . . . . . . . . . . . . . . . . . 1563. Properties of the ring ........................... 1574. Graded prime ideals ............................... 160

. General results on filtered rings and modules . . . . . . . . . . . . . .. Filtered rings and modules ..........................

2. The order function .................................3. The graded module associated with a filtered module . . . .4. Homomorphisms compatible with filtrations . . . . . . . . . . .5. The topology defined by a filtration ..................6. Complete filtered modules ..........................7. Linear compactness properties of complete filtered modules8. The lift of homomorphisms of associated graded modules9. The lift of families of elements of an associated graded

10 . Application: examples of Noetherian rings . . . . . . . . . . . . .. Complete m -adic rings and inverse limits ..............

12. The Hausdorff completion of a filtered module . . . . . . . . .13. The completion of a semi -local ring .........

module ........................................

162162165165169170173176177

179183185187192

CONTENTS CONTENTS

3. m-adic topologies on Noetherian rings . . . . . . . . . . . . . . . . . . . . 195

. Good filtrations 1952. m-adic topologies on Noetherian rings 1993. Zariski rings 2014. The Hausdorff completion of a Noetherian ring . . . . . . . . 2025. The completion of a Zariski ring ..................... 206

4. Lifting in complete rings ................................ 2091. Strongly relatively prime polynomials ................. 2092. Restricted formal power series ........................ 2123. Hensel’s Lemma 2154. Composition of systems of formal power series . . . . . . . . . . 2185. Systems of equations in complete rings . . . . . . . . . . . . . . . . 2206. Application to decompositions of rings . . . . . . . . . . . . . . . . 225

5. Flatness properties of filtered modules ..................... 2261. Ideally Hausdorff modules .......................... 2262. Statement of the flatness criterion .................... 2273. Proof of the flatness criterion 2284. Applications 230

Exercisesfor 1 ........................................... 232

..................................... . . . . . . . . . . . . . . .

......................................

...................................

..............................................................

Exercises for ...........................................Exercises for 3 ...........................................Exercises for 4 ...........................................Exercises for 5 ...........................................

CHAPTER IV . ASSOCIATED PRIME IDEALS AND PRIMARY DECOMPOSITION 261261

261263265

ring 265267267270270

1. Prime ideals associated with a module ....................1. Definition of associated prime ideals. Localization of associated prime ideals . . . . . . . . . . . . . . . .3. Relations with the support

. . . . . . . . . . . . . . . . . ...........................

4. Th e case of finitely generated modules over a Noetherian.............................................................................. Primary decomposition

1. Primary submodules ................................ Th e existence of a primary .............

3. Uniqueness properties in the primary decomposition . . . .4 . The localization of a primary decomposition . . . . . . . . . . .

5. Rings and modules of finite length . . . . . . . . . . . . . . . . . . .

3. Primary decomposition in graded modules . . . . . . . . . . . . . . . .

3. Primary decomposition in graded modules .............Exercises for 1 ...........................................Exercises for 2 ...........................................Exercises for $ 3 ........................................... 301

2746. Primary decomposition and extension of scalars . . . . . . . . 279

2831. Prime ideals associated with a graded module . . . . . . . . . . 283. Primary submodules corresponding to graded prime ideals 284

. INTEGERS ........................................1. Notion of an integral element ...........................

1. Integral elements over a ring ........................

3. Examples of integrally closed domains . . . . . . . . . . . . . . . .

6. Norms and traces of integers

8. Integers over a graded ring

2. The integral closure of a ring . Integrally closed domains

4. Completely integrally closed domains . . . . . . . . . . . . . . . . .5. The integral closure of a ring of fractions . . . . . . . . . . . . . .7. Extension of scalars in an integrally closed algebra . . . . . .9. Application: invariants of a group of automorphisms of an

algebra .........................................

.................................................

303303303308309

312314316318320

. The lift of prime ideals .................................1. Th e first existence theorem ........................... Decomposition group and inertia group ...............

4 . The second existence theorem

1. The normalization lemma

325325330337343

344344

field ............................................ 348349351

........................................... 362

Exercisesfor $ 3 ........................................... 370

. and inertia for integrally closed domains . ........................

...................3. Finitely generated algebras over a field............................ The integral closure of a finitely generated algebra over a

3. The Nullstellensatz .................................. Jacobson rings .....................................

Exercises for ...........................................

CONTENTS CONTENTS

4. Essential valuations of a Krull domain ................. Approximation for essential valuations . . . . . . . . . . . . . . . .

6. Prime ideals of height in a Krull domain ............7. Application : new characterizat ions of discrete valuation

. The integral closure of a Krull domain in a finite extension9. Polynomial rings over a Krull domain ................

482484485

rings ........................................... 487

of its field of fractions ............................ 487488

10. Divisor classes in Krull domains ..................... 489

2. domains ..................................... 4931. Definition of domains ...................... 4932. Characterizations of domains . . . . . . . . . . . . . . . 4943. Decomposition of ideals into products of prime ideals . . . . 4964. The approximation theorem for Dedekind domains . . . . . 4975. The Theorem ......................... 499

3. Factorial domains ..................................... 5021. Definition of factorial domains ....................... 502

2. Characterizations of factorial domains . . . . . . . . . . . . . . . . . 5023. Decomposition into extremal elements . . . . . . . . . . . . . . . . 5044. Rings of fractions of a factorial domain ................ 5055. Polynomial rings over a factorial domain . . . . . . . . . . . . . . 5056. Factorial domains and Zariski rings . . . . . . . . . . . . . . . . . . . 5067. Preliminaries on automorphisms of rings of formal power

series ........................................... 5068. The preparation theorem ........................... 5079. of rings of formal power series ............. 511

4. Modules over integrally closed Noetherian domains ......... 5121. Lattices ........................................... 512. Duality; reflexive modules ........................... 517

.Local construction of reflexive modules

...............521

4. Pseudo -isomorphisms ............................... 5235. Divisors attached to torsion modules .................. 527. Relative invariant of two lattices ..................... 529. Divisor classes attached to finitely generated modules .... 531

a. Properties relative to finite extensions of the ring of scalars 535. A reduction theorem ............................... 54.0

for 1 ........................................... 545

........................................... 556

...................... Modules over Dedekind domains

Exercises for 3 3 ...........................................for 34 ...........................................note (Chapters I to VII) ..........................

Bibliography ............................................Index of notation ..........................................

of terminology ......................................Table of implications .......................................

Table of -11 ..................................Invariances under completion ...............................

....................................Table of invariances -

INTRODUCTION

The questions treated in this Book arose during the development of the theoryof algebraic numbers and (later) algebraic geometry (cf. the Historical Note).From the 19th century onwards these two theories began to show remarkableanalogies; the attempt to solve the problems they posed led to the isolation of anumber of general ideas whose field of application is not limited to rings of algebraic numbers or algebraic functions; and, as always, it is advantageous to

consider these in their most general form in order to see their true significanceand the repercussions of one study on another. The concepts treated in this

INTRODUCTION

ourselves to Noetherian rings) to give for every ideal a lessprecise than a decomposition as a product of prime ideals : here the product is infact replaced by the intersection and the powers of prime ideals by “primary”ideals connected with the prime ideals associated with the ideal in question(bu t which are not direct generalizations of powers of prime ideals). Th e intro -

duction of prime ideals associated with a n ideal and th e study of their proper -ties is the subject of Chapter I V; here also the existence and certain uniquenessproperties of the “primary decompositions” to which we have just alluded areproved; but i t seems at present th at these decompositions usually only play anaccessory role in applications, the essential notion being that of prime idealassociated with an ideal.

In Chapter VII we examine in more detail rings whose properties mostnearly approach those of rings of algebraic integers as far as decomposition as aproduct of prime ideals is concerned; amongst other things it is possible tointroduce into these rings the notion of “divisor”, which is the geometric aspectof this decomposition and plays an important role in algebraic geometry.

Finally Chapters VII I et seq. will deal with notions of more interest in alge -braic geometry than in arithmetic (where they become trivial) and notably theconcept of dimension.

With these notions we come to the frontier of algebraic geometry proper, afrontier which is ever moving and difficult to trace. For, if commutative algebrais an essential tool for the development of algebraic geometry in all its gene -rality, conversely (as has already been seen above) the language of geometryproves very convenient for expressing the theorems of commutative algebraand suggesting a certain intuition naturally enough absent from abstractalgebra; with the tendency to enlarge more and more the limits of algebraicgeometry, algebraic and geometric language tend more than ever to merge.

CHAPTER I(*)

Flat Modules

otherwise stated, all the rings considered in this chapter are assumed to have a unit ring homomorphisms are assumed to map unit element to unit element. A

subringof a ring A a containing the unit element of A.a ring, M a left A-module and U (resp. V) an additive subgroup (resp.

M ) ,recall that UV or U is used to denote the additive subgroup generated by theuv, where u U , v V (Algebra,Chapter VIII, 6 , no. 1). a is anwe write = A. For any set E, 1, (or 1 no confusion arises)is used to

denote the identity mappingof E to itself. Recall that the fo r a module imply that E is a (resp. right) module over

a A and 1 denotes the unit element o f A, then 1 = x (resp. x . = x ) fo rE (Algebra,Chapter $1, no. 1). and F are two (resp. right)

recall that F) (or simply F)) is used to denote thegroup of o f E to F 1, no. 2). B y an abuseo f

will often be used to denote a module reduced to its identity element.

1. DIAGRAMS AND EX ACT SEQUENCES

1. DIAGRAMS

Let, for example, A, B, C , D, E be five sets and let f be a mapping from A to B,g a mapping from B to C, h a mapping from D to E, u a mapping from B to Dand a mapping from C to E. To summarize such a situation we often make use

diagrams; for example, the above situation is summarized by the following

With the exception of 4, the results of this chapter depend only on Books I

CHAPTER

Graduations, Filtrations

and Topologies

All the rings considered in this chapter are assumed to have a unit element; all theare assumed to map unit element to unit element. By a subring a ring

A we mean a containing the unit element of A. Unless otherwise stated, all themodules are modules.

FINITELY GENERATED GRADED ALGEBRAS

1. SYSTEMS OF GENERATORS OF A COMMUTATIVE ALGEBRA

Let A be a commutative ring and B a commutative A -algebra. Let us recall(Algebra, Chapter IV, 2, no. 1) that if = is a family of elements

B, the mapping f f ( x ) from the polynomial algebra to ishomomorphism of onto the subalgebra of B generated by the x, ,whose kernel a is the ideal of polynomials f such that f ( x ) = 0, called the ideal

algebraic relations (with in A) between thex,.

DEFINITI ON 1. In a commutative algebra B over a commutative ring A, aofelements is called algebraicallyfree over A (or the x, are called algebrai -

independent over A) ideal ofalgebraic relations between thex,, withA, is reduced to 0. A famil y which is not algebraically free over A also called

algebraically related (or its elements are also called algebraically dependent over A) .

definition generalizes that given in Algebra, Chapter V, 5, no. 1, Defini -tion 1 for families of elements of a commutative field.

Except in 5, which uses the results of Chapter I, 4 and therefore homo -logical algebra, no use is made in thi s chapter of any books other than I to VI

Chapters I and of this book.

CHAPTER V(*)

Integers

Unless otherwise stated,all the rings and all the algebras considered in this chapter areassumed to be commutatiue and to possessa unit element; all the ring homomorphisms areassumed to map unit element to unit element. B ya subring of a ring A we meana subringcontaining the unit element of A.

to the intersection of the maximal ideals of A containing a, which is the radicalof a (Corollary to Proposition 3).COROLLARY 3. Let A be a Jacobson ring. there exists generated A-algebraB containingA and whichis afield, thenA is and B is a n algebraicextension

suffices to apply Theorem 3 (ii) with m' = (0)

EXERCISES

suffices to apply Theorem 3 (ii) with m (0).

1. Let d be a rational integer not divisible by a square in Z. Show that theelements of the field K = which are integral over Z are of the form

+ where a b if d - 1 0 (mod. 4) and the elements( a + where a Z, b Z, a and b both even or a and b both odd, if

d - 1 0 (mod. 4) (cf. Algeb ra, Chapter VII , 1, Exercise 8). Deduce tha tA = is not integrally closed and give an example of an element of

which is integral over A bu t whose minimal polynomial over the field

of fractions of A does not have coefficients in A.2. Let K be a commutative field and A the sub -K-algebra of the polynomial

algebra generated by th e monomials (k Show tha t XYis such that is contained in a finitely generated A -module but XY isnot integral over A.

3. Give an example of an infinite sequence (K,) of extensions of a commuta-

K, is nottive field K of finite degree over K such that the K-algebraalgebraic over K.

4. In the matrix ring give an example of two elements integral overbut neither the sum nor the product of which is integral over Z (consider

matrices of the form + where is nilpotent).

5. Let A be a commutative ring, B a commutative ring containing A andan invertible element of B. Show that every element of n is integralover A. (I f = . .+ = . . where the a, and arein A, show that the sub -A-module of B generated by 1, x, . . ., is afaithful -module.)

6. Let A be a commutative ring and B a commutative A-algebra; suppose

354 355

CHAPTER VI

Valuations

Unless otherwise stated, all the rings considered in this chapter are assumed to be com-mutative and to possess a unit element. All the ring homomorphisms are assumed to mapunit element to unit element. Every subringof a ring A is assumed to contain the unit element of A.

CHAPTER VII

Divisors

All the rings in this chapter are assumed to be commutative and to possess a unit element. All the ring are assumed to map unit element to unit element. Every of a ring A is to contain the unit element of A.

1. KRULL DOMAINS

1. DMSORIAL IDEALS OF AN INTEGRAL DOMAIN

DEFI N ITIO N 1. Let A be an integral domain and K of fractions. Everysub - A-modulea such that there exists an element d 0 in A fo r which A is called a

fractional ideal of A (or of K, by an abuse of language).

Every finitely generated sub -A-module a of K is a fractional ideal: for if is a system of generators of a, we may write a, = where A,

A and d, 0 ; if d = . . .d,,, clearly A. In particular the monogenoussub-A-modules of K are fractional ideals (recall that they have been called

fractional principal ideals in Algebra, Chapter VI, 1, no. 5). If A is Noetherian,every fractional ideal is afinite ly generated A-module. Every sub -A-module of afractional ideal of A is a fractional ideal. Every ideal of A is a fractionalideal; to avoid confusion, these will also be called the integral ideals of A.

We denote by the set of non-zero fractional ideals of A. Given two ele -

ments a, b of we shall write a b (or b a) for the relation “everyfractional principal ideal containing a also contains clearly this relation is aon Let R denote the associated equivalence relation b and

a” (Set Theory,Chapter 1, no. 2) and the quotient setwe shall say that the elements of are the divisors of A and, for everyfractional ideal a we shall denote by div a (or div, a) the canonicalimage of a in and we shall say that div a is the divisor of a; if a = A x is a

for every ideal in these rings (*).He does not seem to be concerned with ques-tions of uniquenessin this decomposition; it is Macaulay who, a little laterintroduces the distinction between “immersed” and primaryideals and shows that the latter a re determined uniquely, bu t not the former. Itshould finally be noted that Lasker also extends his results to the ring of convergent

power series in a neighbourhood of a point, by using Weierstrass’s “preparationThis par t of his memoir is no doubt the first place this ring had been

considered from a purely algebraic point and the methods which Laskerdevelops on this occasion were strongly to influence Krull when in 1938 hecreated the general theory of local rings p. 204 and passim).

* * *The movement of ideas which will give birth to modern Commutative Algebrabegins to take shape abou t 1910. If the general notion of field was reached bythe beginning of the 20th century, in contrast the first work where the general

f d f d b bl h f k l h

HISTORICAL NOTE

with negative exponent). shows similarly that, if is a prime ideal of k lying above a prime a “p -adic series” may be associated with every

k, of the form (or when is ramified over the beingtaken in a given representative system of the field of residues of the ideal buthis great originality lies in having had the idea of considering such “expansions”even when they correspond to no element of k, by analogy with the expansions inintegral series of transcendental functions on a Riemann surface

Throughout the rest of his career, Hensel devotes himself to polishing andperfecting little by little his new calculus; and if his manner seems to us hesitantor ponderous, it must not be forgotten that at the beginning at least he hadat his disposal none of the topological or algebraic tools of modern mathematicswhich would have facilitated his task. In his first publications he moreoverscarcely speaks of topological notions and on the whole for him the ring of adic integers a prime ideal in the ring of integers A of a number field k ) is,in modern terms, the inverse limit of the rings for n increasing indefinitely,i l l b i d bli h h i f hi i d i

notion of ring is defined is probably that of Fraenkel in 1914 At this time,there were already as examples of rings, not only th e integral domains of theTheory of Numbers and Algebraic Geometry, but also rings of power series(formal and convergent) and finally algebras (commutative or not) over abase field. However, for the theory of fields as well as that of rings, the catalystrole seems to have been played by Hensel’s theory of numbers, whichFraenkel and also Steinitz mention specially as the starting point of theirresearch.

Hensel’s first publication on this subject goes back to 1897; he there startsfrom the analogy shown by Dedekind and between the points of a

Riemann surface of an algebraic function field K and the prime ideals of anumber field he proposes to carry over to the Theory of Numbers “Puiseuxexpansions” (classical from the middle of the 19th century) which, in abourhood of any point of the Riemann surface of K, allow every element

to be expressed in the form of a convergent series of powers of theat the point considered (a series with only a finite number of terms

zeros of a’ has only irreducible components of dimension - which allowshim to conclude by induction.

(*) I t interesting to note that second proof of the unique decom -position theorem proceeds by first establishing the existence of a unique reducedPrimary decomposition; and in a passage not published in the 1 lt h supplement,

observes explicitly that this part of the proof is valid not only forA of all integers of a number field K, but also for all the “orders” of

303). I t is only then, after showing explicitly that A is “completely inte -grally closed” (to within terminology) that h e proves, using this fact, thatprimary ideals of the above decomposition are in fa ct powers of prime

vol. p. 307).

in a purely algebraic sense; and to establish the properties of this ring and itsfield of fractions, it is necessary at each step to use more or less painfully ad arguments (for example to prove that the p -adic integers form an integraldomain). Th e idea of introducing topological notions into a p-adic field doesnot appear in Hensel’s works before 1905 and it is only in 1907,having published the book where he reexpounds the theory of algebraicnumbers according to his ideas that he arrives at the definition andessential properties of absolute values starting with which he willbe able to develop, modeling it on Cauchy’s theory, a new “p-adic analysis”which he will be able to apply fruitfully in the Theory of Numbers (notably byusing the p-adic exponential and logarithm) and whose importance has beengrowing ever since.

Hensel had well seen, from the beginning, the simplifications his theorybrought to classical expositions, by allowing the problems to beand the work to be carried out in a field where not only are the divisibilityproperties trivial, but also, thanks to the fundamental lemma which he dis-covered as early as 1902 the study of polynomials whose “reduced”polynomials modp have no multiple roots is reduced to the study of poly -nomials over a finite field. He had given as early as 1897 striking exam-ples of these simplifications, notably on questions related to the discriminant(in particular, a short proof of the criterion he had given a few years earlier forthe existence of “extraordinary divisors”). But for a long time it seems thatthe p -adic numbers inspired considerable distrust in contemporary mathe -

maticians; a current attitude no doubt towards ideas that a re “too abstract”,but which was also justified in p art by the rather excessive enthusiasm of theirauthor (so frequent in mathematics among zealots of new theories). Not con -tent to apply his theory fruitfully to algebraic numbers, Hensel, impressed

HISTORICAL

as all his contemporaries were, by the proofs of the transcendence of e andand perhaps misled by the adjective “transcendental” applied both to

to functions, had come to think that there existed a connectionbetween his p -adic numbers and transcendental real numbers and he hadthought for a moment that he had obtained a simple proof of the transcen-dence of and even of ([ p. 556) (*).

Soon after 1910, the situation changes, with the rising of the next generation,influenced by the ideas of Frtchet and F. on topology and by those of Steinitz on algebra, and from the start devoted to “abstraction”; it willknow how to assimilate and put in their true place works. As early as1913, gives a general definition of the notion of absolute value,recognizes the importance of ultrametric absolute values (of which the p -adic

absolute value was an example), proves (by modelling the proof on the case of real numbers) the existence of the completion of a field with respect to anabsolute value and above all shows generally the possibility of extending anabsolute value to any algebraic extension of the given field. But he had

type of ring (Artinian with only a single prime ideal, which is moreoverassumed to be pr incipal). With the exception of Steinitz’s work on fieldsthe first important works on the study of general commutative rings are E.Noether’s two great memoirs on ideal theory : that of 1921 devoted toprimary decomposition, which takes up again in all generality and completeson many points the results of Lasker and Macaulay; and tha t of 1927 charac -terizing Dedekind domains axiomatically Just as Steinitz had shown forfields, it is seen in these memoirs how a small number of abstract ideas, such asthe notion of irreducible ideal, the chain conditions and the idea of an inte -grally closed domain (the last two, as we have seen, already brough t to light byDedekind) can by themselves lead to general results which seemed inextricablybound up with results of pure computation in the cases where they hadpreviously been known.

With these memoirs of E. Noether, joined to the slightly later worksvan der Waerden on divisorial ideals and Krull relating these ideals toessential valuations the long study of the decomposition of ideals

d l (*) h l h d

y g gnot seen that the ultrametric character of an absolute value was alreadyrevealed in the prime field; this point was established by to whomalso is due the determination of all the absolute values on the field Q and thefundamental theorem characterizing fields with a non -ultrametric absolutevalue as subfields of C In the years from 1920 to 1935, the theory will becompleted by a more detailed study of absolute values which ar e not necessarilydiscrete, including amongst others the examination of various circumstanceswhich arise in passing to an algebraic o r transcendental extension (Ostrowski,Deuring, F. K. Schmidt); on the other hand, in 1931, Krull introduces andstudies the general notion of valuation which will be greatly used in theyears that follow by Zariski and his school of Algebraic Geometry Wemust also mention here, although it lies outside our scope, the deeper studieson the structure of complete valued fields and complete local rings, which datefrom the same period (Hasse -Schmidt, Witt, Teichmuller, I. Cohen).

* * *The work of Fraenkel mentioned above (p. 594) only treated a very special

(*) This research at all cost of a narrow parallelism between p -adic series andTaylor series also leads Hensel to pose himself strange problems: he proves for

example that every p-adic integer may be written in the form of a series k= O

where the are rational numbers chosen so that the series converges not onlybut also in R (perhaps by analogy with Taylor series which converge

several places at once ?) ([ and [ .An example of a valuation of height 2 had already been introduced inciden -

tally by H. Jung in 1925

started a century earlier (*) is thus complete, at the same time as modernCommutative Algebra is being inaugurated.

Th e innumerable later research works on Commutative Algebra are groupedmost easily according to several importan t directions of development :

(A) Local rings and topologies

Although the germ was contained in all the earlier works on the Theory of Numbers and Algebraic Geometry, the general idea of localization came tolight very slowly. The general notion of ring of fractions is only defined in

1926 by H. Grell, a pupil of E. Noether, and only for integral domainsits extension to more general rings will only be given in 1944 by C. Chevalleyfor Noetherian rings and in 1948 by Uzkov in the general case. Until about1940, Krull and his school are practically alone in using in general argumentsthe consideration of the local rings A, of an integral domain A ; these rings willonly begin to appear explicitly in Algebraic Geometry with the works of valley and Zariski starting in 1940

Th e general study of local rings themselves only begins in 1938 with Krull’s

(*) Following the definition of divisorial ideals, a considerable number of research works (Priifer, Krull, etc.) were undertaken on ideals whichare invariant under other operations a a‘ satisfying axiomatic conditions analo-gous to the properties of the operation a A: (A: a) which gives birth to divisorialideals; the results obtained in this way have as yet found no application in AlgebraicGeometry nor in the Theory of Numbers.

I n the works of Hensel and his pupils on the Theory of Numbers, the localrings A, are systematically neglected to the benefit of their completions, nobecause of the possibility of applying Hensel’s lemma to the latter.

HISTORICAL NOTE

theorems on the lifting of prime ideals to integral algebras as also forextending the theory of decomposition and inertia groups of Dedekind -Hilbert As for E. Noether, we owe to her the general formulation of thenormalization lemma (*) (from which follows amongst other things Hilbert’sNullstellensatz) as also th e first general criterion (transcribing the classicalarguments of Kronecker and Dedekind) for the integral closure of an integraldomain to over that domain.

Finally, it should be pointed out here th at one of the reasons for the presentimportance of the notion of integrally closed domain is due tostudies on algebraic varieties; he discovered that varieties (that isthose whose local rings are integrally closed domains) are distinguished byparticularly pleasant properties, notably the fact tha t they have no “singularity

and it has then been seen tha t analogous phenomena are truefor “analytic spaces”. Therefore (that is the operation which,for the local rings of a variety, consists of taking their integral closures) hasbecome a powerful weapon in the arsenal of modern Algebraic Geometry.

HISTORICAL NOTE

Conversely, it could be foreseen tha t the new classes of modules introducednaturally by Homological Algebra as “universal of the Extfunctors (projective modules and injective modules) and the Tor functors(flat modules) would throw new light on Commutative Algebra. It happenstha t chiefly projective modules and still more flat modules have shown them -selves useful: the importance of the latter arises above all from the remark,made first by Serre that localization and completion introduce flatmodules naturally, thus “explaining” in a much more satisfactory way theproperties of these operations already known and rendering them mucheasier to use. I t should moreover be mentioned (as we shall see in later chapters)that the applications of Homological Algebra are far from being limited tothis and that it is playing a more and more important role in AlgebraicGeometry.

(E) The notion of spectrumThe most recent in date of the new notions of Commutative Algebra has a com -plex history Hilbert’s spectral theorem introduced ordered sets of orthogonal

(D) The study of modules and the of Homological Algebra

One of the striking characteristics of the work of E. Noether an d W. Krull inAlgebra is the tendency to “linearization”, extending the analogous develop -ment given to field theory by Dedekind and in other words, ideals areconsidered above all a s modules and so all the constructions of Linea r Algebra(quotient, product and more recently tensor product and formation of homo -morphism modules) are brought to bear on them, producing i n general moduleswhich are no longer ideals. It is thus quickly seen that in many questions (for

commutative or non -commutative rings), interest should not be confined to thestudy of ideals of a ring A, but on the contrary the theorems should be statedin general for A -modules (sometimes subjected to cert ain finiteness conditions).

The intervention of Homological Algebra has strongly reinforced the abovetendency, since this branch of Algebra is essentially concerned with questionsof a linear nature. We shall not retrace its history here; but it is interesting topoint ou t tha t several fundamental notions of Homological Algebra (such asthat of projective module and that of Tor functor) came into being on theoccasion of a close study of the behaviour of modules over a Dedekind domainrelative to the tensor product, a study undertaken by H. in 1948.

on this subject is that where I. Cohen and Seidenberg extend Krull’s liftingtheorems, indicating exactly the limits of their validity It should be mentioned

that E. Noether had explicitly mentioned the possibility of such generalizations inher memoir of 1927 p. 30).

(*) A particular case had already been asserted by Hilbert in 1893p. 316).

plex history. Hilbert s spectral theorem introduced o rdered sets of orthogonalprojectors of a Hilbert space, forming a “Boolean (or rather aBoolean lattice) in one -to-one correspondence with a Boolean lattice of classes of measurable subsets (for a suitable measure) of R. No doub t his earlierwork on operators on Hilbert spaces, about 1935, led M. H. Stone to studyBoolean lattices generally and notably to look for of them bysubsets of a set (or classes of subsets with respect to a certain equivalence rela -tion). H e observes tha t a Boolean lattice becomes a ring (moreoverof a very special type), if multiplication is defined on i t by xy = inf (x, y) andaddition by x +y = y)). In the particular case where the

Boolean lattice in is the set of all subsets of afinite set X, it isimmediately seen that the elements of X are in a natural one -to-one corres -pondence with the maximal ideals of the corresponding ring; andStone obtains precisely his general representation theorem for a Boolean latticeby similarly considering the set of maximal ideals of the corresponding ring a ndassociating with each element of the Boolean lattice the set of maximal idealswhich contains it

On the other hand, the set of both open and closed subsets of a topologicalspace was a well -known classical example of a Boolean lattice. In a secondpaper Stone showed that in fact every Boolean lattice is also isomorphicto a Boolean lattice of this nature. For this it was of course necessary to define atopology on the set of maximal ideals of a “Boolean” ring; which was very

(*) A Boolean lattice is a lattice -ordered set E, with a least element a and agreatest element where each of the laws sup and inf is distri butiv ewith respect tothe other and, for all a E, there exists a unique a‘ E such that a’) aan d a’) = w (cf. Set Theory, Chapter 1, Exercise 17).

INDEX OF TERMINOLOGY

The reference numbers indicate the chapter, section and sub -section (orexercise) in that order.

Canonical decomposition of a place: VI.2.3- factorization of a valuation: VI.3.2- homomorphism of the decomposition group of a prime ideal of A' into

Class, divisor, attached to a finitely generated module :Classes, divisor of) : VII.1.2Closure, algebraic, of a field in an algebra : V. 1.2- integral, of an integral domain : V. 1.2- integral, of a ring in an algebra: V.1.2Commutative diagram: I. 1.2Compatible (filtration) with a ring structure, module structure: 111.2.1- (homomorphism) with filtrations: 111.2.4Complete system of extensions of a valuation: VI.8.2Completely integrally closed domain : V. 1.4Component, irreducible (of a topological space) : 11.4.1Conditions, Hensel's: 111.4.5Conductor of a submodule : V 1 5

the automorphism group of

restricted, principal restricted adtle :m-adic filtration: 111.2.1

n-adic integers: 111.2.12Algebra, Azumaya:- finitely generated : 111.1.1- integral, finite, over a ring: V. 1.1Algebraic closure of a field in an algebra : V. 1.2Algebraically closed field in a n algebra: V. 1.2- dependent, independent, elements: 111.1.1- free, related, family: 111.1.1Almost all (property valid for) : VII.4.3- nilpotent endomorphism: IV. 1.4Approximation theorem for absolute values : VI.7.3- theorem for valuations: VI.7.2

Lemma: 111.3.1Associated (filtered module) with a graded module: 111.2.1- (filtered ring) with a graded ring: 111.2.1- (filtration) with a graduation:

111.2.5

- (graded with a homomorphism compatible with filtra-tions: 111.2.4

- (graded module) with a filtered module: 111.2.3

- (graded ring) with a filtered ring: 111.2.3- (mapping) with a ring homomorphism: 11.4.3- (prime ideal) with a module: IV. 1.1- ring, place, valuation: VI.3.3

Conductor of a submodule : V. 1.5Content of a polynomial over a pseudo-Bezoutian domain: V II. 1- of a torsion module: VII.4.5Criterion, irreducibility:

Decomposition, canonical, of a place : VI.2.3- complete, of a prime ideal: V.2.2- field of a prime ideal: V.2.2- group, ring, of a prime ideal: V.2.2- of an ideal in a Dedekind domain into prime factors: VII.2.3- primary: IV.2.2 and- reduced primary: IV.2.3 andDecreasing filtration: 111.2.1Dedekind domain: VII.2.1Defined (topology) by a filtration: 111.2.5Defining ideal:Degree, residue class, of one valuation over another:Dependence, integral (equation of) : V. 1.1Derived (module filtration) from a ring filtration: 111.2.1Diagram, commutative: I. 1.2- snake : I. 1.4Discrete filtration: 111.2.1- valuation: VI.3.6

Distinguished polynomial : VII.3.8Divisor, principal divisor: VII.l. 1-- finitely generated:

Divisorial fractional ideal : VII. 1.1Divisors, equivalent: VII . 1.2Domain, Bezoutian (o r Bezout) :- completely integrally closed : V. 1.4- Dedekind: VII.2.1- factorial: VII.3.1- integrally closed : V. 1.2- integrally closed, of finite character: VII. 26 and 28- integrally Noetherian:

Krull: VII.1.3- local integral, of dimension 1:- Pruferian (or Prufer):- pseudo-Bezoutian: VII . 1.Ex.21- pseudo -principal: VII. 1

- pseudo-Pruferian: 19- regularly integrally closed: VII. 1Dominating (local ring) a local ring: VI . 1.1Dual algebraic toric of a module:

Field, residue, of a place: VI.2.3- residue, of a valuation: VI.3.2- value, of a place: VI.2.2Filtered group, ring, module: 111.2.1Filtration, m-adic: 111.2.1- associated with a graduation: 111.2.1- compatible with a ring structure, module structure: 111.2.1- discrete: 111.2.5- m-good: 111.3.1- increasing, decreasing, separated, exhaustive : 111.2.1- induced, product, quotient:- module, derived from a ring filtration:- trivial : 111.2

Finite algebra over a ring : V. 1.1- (place) a t an element: VI.2.2Finitely generated algebra: 111.1.1Finitely presented: 1.2.8Fl f M ( d l ) 1 2 2

Dual, algebraic toric, of a module:- lattice: VII.4.2- topological toric:

Element, topologically nilpotent: 111.4.3Elements, algebraically dependent, independent: I I I. 1.- strongly relatively prime : 111.4.1Endomorphism, almost nilpo tent: IV. 1.4Equivalent divisors: VI I. 1.2- valuations: VI.3.2Essential graded ideal:- valuations: VII.1.4Euclidean ordered field:Exhaustive filtration:Extension, quasi -Galois: V.2.2

Factor, invariant: 11 and 14Factorial domain: VII.3.1Factorization, canonical, of a valuation : VI.3.2Faithfully flat module: 1.3.1Family, algebraically free, related : 111.1.1- formally free: 111.2.9Field, algebraically closed in an algebra : V. 1.2

- decomposition: V.2.3- projective: VI.2.1- residue, of a local ring:

Flat for M, (module) : 1.2.2- module: 1.2.3Formally free family: 111.2.9Fractional ideal:Function, order: 111.2.2

Gaussian integer: V. 1.1Gauss’s lemma: VII.3.5Gelfand -Mazur Theorem : V1.6.4.

Generated by a subset (multiplicative subset) : 11.2.1Generators, formal system of: 111.2.9m-good filtration: 111.3.1Group, decomposition : V.2.2- filtered: 111.2.1- inertia : V.2.2- of classes of invertib le modules: 11.5.7- of operators, locally finite : V. 1.9- operating on a ring: V. 1.9- order, of a valuation: VI.3.2- ordered, of height n, of height + : VI.4.4

Height 1 (prime ideal of) : VII. 1.6- of an ordered group, of a valuation : VI.4.4Henselian ring :Hensel’s conditions: 111.4.5- Theorem: 111.4.3

Invertible fractional ideal: 11.5.7- submodule: 11.5.6Irreducible component: 11.4.1- set: 11.4.1- space: 11.4.1Isolated subgroup: VI.4.2

Jacobson ring: V.3.4

Homomorphism, canonical, of the decomposition group of a prime ideal- from A' to the automorphism group of V.2.2- compatible with filtrations:- graded, associated with a homomorphism compatible with filtrations :

- local: 11.3.1111.2.4

- pseudo -injective, pseudo -surjective, pseudo -zero, pseudo -bijective :VII.4.4

Ideal, determinantal: and 14- essential graded :- immersed prime: IV.2.3- integral, fractional ideal : VII. 1.1- invertible fractional: 11.5.7- lying above an ideal : V.2.1- minimal prime: 11.2.6- of a place: VI.2.3

p- of a valuation: VI.3.2- prime:- prime, associated with a module : IV. 1

- prime, decomposing completely: V.2.2- prime, of height 1: VII . 1.6- primary, p -primary: IV.2.1 and- unramified: V.2.Ex.18 and 19Ideally Hausdorff module: 111.5.1Ideals, relatively prime: 11.1.2

Identity, Cauchy's:Immersed prime ideal: IV.2.3Improper valuation: VI.3.1Increasing filtration:Independent valuation rings: VI.7.2- valuations: VI.7.2Index, initial, of a subgroup of an ordered group, initial ramification index

valuation: VI.8.4- ramification: VI.8.1Induced filtration: 111.2.1Induction, Noetherian (principle of) :Inertia field: V.2.3- ring, group: V.2.2Initial ramification index: VI.8.4Integer, algebraic: V. 1.1- Gaussian : V. 1.1- over a ring: V.l.l

Module graded, associated with a filtered module: 111.2.3- ideally Hausdorff 111.5.1- of fractions defined by a subset of a ring: 11.2.2- projective, of rank n: 11.5.3- pseudo -coherent:- pseudo -zero: VII.4.4

divisor class : VII . 1.2Morphism for laws of composition not everywhere defined: VI.2.1Multiplicative subset: 11.2.1

Nilradical of a ring: 11.2.6Noetherian space: 11.4.2

Non -degenerate submodule : 11.5.5Normalization lemma: V.3.1Normed discrete valuation:Nullstellensatz: V.3.3

Pseudo -injective, pseudo -surjective, pseudo -zero, pseudo -bijective (homomor -

Pseudo -isomorphism: VII.4.4Pseudo -zero module: VII.4.4

phism) : VII.4.4

Quasi -Galois extension: V.2.2Quotient filtration:

Radical of an ideal: 11.2.6Rank at p of a projective module: 11.5.3- of a projective module: 11.5.3- rational, of a commutative group : VI. 10.2-

residue: VI.8.5Rational rank of a commutative group:Reduced order: VII.3.8- primary decomposition: IV.2.3- ring:11 2 6

Order group of a valuation: VI.3.2- of an element for a valuation: VI.3.2- reduced, of a formal power series: VII.3.8Ordered pair of rings with the linear extension property: 1.3.7

theorem: VI.6.4

Place, finite at x : VI.2.2- of a field: VI.2.2- trivial: VI.2.2Point, generic, of an irreducible space:Polygon, Newton :Polynomial, distinguished: VII.3.8- minimal : V. 1.3Preparation theorem: VII.3.8Presentation of a module, - finite: 1.2.8n-presentation:Presented, finitely (module) : 1.2.8Primary decomposition: IV.2.2 and- ideal, submodule: IV.2.1 andPrime ideal:- spectrum: 11.4.3Principal divisor: VII. 1.1

- restricted VII.2.4Principle of Noetherian induction: 11.4.2Product filtration:Projective field: VI.2.1

ring: 11.2.6- series: VII.3.8Reflexive lattice: VII.4.2Related local rings:Relatively prime ideals:Representative system of extremal elements: VII.3.3Residue class degree of a valuation: VI.8.1- - field: 11.3.1- rank of a valuation:Resolution, finite free, of a module: VII.4.7Restricted VII.2.4- formal power series: 111.4.2Ring, absolutely flat:- coherent (left, right ) :- decomposition: V.2.2- filtered: 111.2.1- filtered, associated with a graded ring: 111.2.1- graded, associated with a filtered ring: 111.2.3- inertia: V.2.2- integrally closed in an algebra : V. 1.2- Jacobson: V.3.4- linearly topologized: 111.4.2- local: 11.3.1

- local, dominating a local ring: V I. 1.1- local, of A at p, of p (p a prime ideal) : 11.3.1- of a place: VI.2.3- of a valuation: VI.3.2

Ring of fractions defined by a subset of a ring : 11.2.1- reduced: 11.2.6semi-local: 11.3.5total, of fractions: 11.2.1unramified:

- valuation, valuation ring of a field : VI. 1.1- Zariski: 111.3.3Rings, independent valuation: VI.7.2

Saturation of a submodule with respect to a multiplicative subset (with respect

Semi-local ring: 11.3.5

Series, reduced: VI I.3.8- restricted formal power: 111.4.2Set, irreducible: 11.4.1- major, in a totally ordered group: VI.3.5Space irreducible: 11 4 1

to a prime ideal) : 11.2.4

Theorem, preparation: VII.3.8- Stickelberger's : 18- Zariski's Principal:Topologically nilpotent element: 111.4.3Topology defined by a filtration: 111.2.5- spectral: 11.4.3- Zariski: 11.4.3Transporter: 1.2.10Trivial filtration: 111.2.1- place: VI.2.2

Ultrametric absolute value:Uniformizer for a discrete valuation:Unramified valuation: VI.8.1

Valuation, valuation of an element x : VI.3.1 and 2- discrete, normed discrete valuation: VI.3.6-

Space, irreducible: 11.4.1- Noetherian: 11.4.2Special topology: 11.4.3Spectrum, prime, of a ring: 11.4.3Strongly Laskerian module:- primary submodule:- relatively prime elements: 111.4.1Subgroup, isolated, of an ordered group: VI.4.2Submodule, invertible :- non-degenerate: 11.5.5- primary, p-primary: IV.2.1Subset, multiplicative, of a ring: 11.2.1- multiplicative, generated by a subset: 11.2.1- saturated multiplicative:Support of a module: 11.4.4System, complete, of extensions of a valuation: VI.8.2- formal, of generators: 111.2.9- representative, of extremal elements: VII.3.3

Theorem, approximation, for absolute values: VI.7.3- approximation, for valuations: VI.7.2- Gelfand-Mazur: VI.6.4--Hilbert's zeros (Nullstellensatz) : V.3.3- 111.3.1- VII.3.5- Ostrowski's: VI.6.3

,- essential: VII.1.4 and Ex.26- improper: VI.3.1- ring: VI.l.l- unramified: VI.8.1equivalent: VI.3.2

- independent: VI.7.2Value field of a place: VI.2.2- ultrametric absolute: VI.6.1

Weakly associated (prime ideal) with a module: IV. 17

Zariski, ring: 111.3.3- 11.4.3

TABLE OF IMPLICATIONS

semi -local ring

local ring

integrally closed domain valuation ring

completely integrally valuation ringclosed domain of height

domain

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