A Galois theory of commutative rings - CORE · satisfy Galois correspondence theorems which support...

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e Galoisnce of, vol. 3,(1983)

ial cor-biringsof Ga-er, Thee fromsimple

alge-ence

pired

Journal of Algebra 289 (2005) 380–411

www.elsevier.com/locate/jalgebr

A Galois theory of commutative rings

David J. Winter

University of Michigan, Ann Arbor, MI 48109-1109, USA

Received 23 July 2004

Available online 22 April 2005

Communicated by Michel Van den Bergh

Abstract

Galois objects—Galois groups, rings, Lie rings, and biringsG —act on commutative ringsA andsatisfy Galois correspondence theorems which support Galois descent. This generalizes ththeory of fields to a Galois theory of commutative rings. In particular, the classical correspondeGalois, the Jacobson–Bourbaki correspondence [N. Jacobson, Lectures in Abstract AlgebraVan Nostrand, 1964; D.J. Winter, The Jacobson descent theorem, Pacific J. Math. 104 (2)495–496; D.J. Winter, The Structure of Fields, Springer-Verlag, 1974], the Jacobson differentrespondence [N. Jacobson, op. cit.; D.J. Winter, The Structure of Fields, op. cit.], the Galoiscorrespondence of [D.J. Winter, The Structure of Fields, op. cit.], and corresponding theorieslois descent [N. Jacobson, Forms of algebras, Yeshiva Sci. Confs. 7 (1966) 41–71; D.J. WintJacobson descent theorem, op. cit.; D.J. Winter, The Structure of Fields, op. cit.] generalizfields to commutative rings. The Galois Lie rings correspondence Theorem 4.2 solves therestricted irreducible derivation rings Problem 8.4 in the finitely generated case. 2005 Elsevier Inc. All rights reserved.

1. Introduction

This paper was inspired by ideas used in building infinite-dimensional simple Liebras [21] which led to a particularly simple proof of Jacobson’s differential correspondtheorem—written down in the foundations paper [19]. In turn, [19] and this proof ins

E-mail address:[email protected].

0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2005.03.012

D.J. Winter / Journal of Algebra 289 (2005) 380–411 381

Jacob-

encetheory

Galoiss

roups,eeneralply to

sn-

ichach

Galois

r

the philosophy of this paper and the proof of its Theorem 4.2 as a generalization ofson’s differential correspondence theorem from fields to commutative rings.

The unifying theme of [19]—using “dual bases” to prove Galois correspondtheorems—continues on as the unifying theme of this generalization of the Galoisof fields to a Galois theory of commutative rings.

The most basic notion underlying this paper is that of aGalois correspondence[19]between setsF andG partially ordered by inclusion—a pair(op,po) of inclusion reversingmapsop: F → G, F ← G : po such that

po(G) ⊇ F ⇔ G ⊆ op(F ) (F ∈ F, G ∈ G).

The notion of a Galois correspondence is quite general—and the condition for acorrespondence may be read as follows where theF andG are referred to, generally, apositionsandoperators:

The position ofG contains the positionF if and only if the operatorG is contained inthe operator ofF .

The roles of the positions (e.g., subfields considered below) and operators (e.g., grings, Lie rings, birings considered below) are reversed in thedual Galois correspondenc(po,op). The dual is a symmetry of the concept of Galois correspondence, so once gdefinitions and theorems are in place for positions (respectively operators), they apoperators (respectively positions) by duality.

An elementF ∈ F (respectivelyG ∈ G) is closedwhen it equals itsGalois correspon-dence closureF ≡ po◦op(F ) (respectivelyG ≡ op◦po(G)). LetF (respectivelyG) denotethe set of closedF ∈ F (respectively closedG ∈ G)—and letop,po denote the restrictionof op, po to F, G, respectively. The functionsop, po are then inverses of each other. Cosequently,(op,po)—called theclosed correspondencedefined by(op,po)—is abijectiveGalois correspondence betweenF,G.

A Galois correspondence theoremis a theorem within some ambientGalois theorywhich establishes that some such(op,po) is a bijective Galois correspondence—or whdescribes someclosedF ’s andG’s of a Galois correspondence which correspond to eother. Such a Galois correspondence theorem is sometimes also referred to as afundamen-tal theoremof that ambient Galois theory.

The fundamental theorem of classical Galois theory establishes that the classicalcorrespondenceG ≡ AutF K , F ≡ KG between the setFG of subfieldsF of a fieldK suchthatK is finite-dimensional GaloisoverF , that is,K is the splitting field overF of someseparable polynomial, and the setG of finite groupsG of automorphisms ofK is bijective.Here,AutF K is the group of automorphisms ofK which fix the elements ofF andKG isthe fixed field ofG onK .

Since a fieldK has no ideals other than 0 andK , the following question arises:

How does Galois theory generalize upon passage from a fieldK to a commutative ringA with respect to Galois objects actingirreducibly on A—without stable ideals othe
than0 andA?

382 D.J. Winter / Journal of Algebra 289 (2005) 380–411

rings,rings,

ively.

rtinghich

-

more

bir-s

rings,to

rmin-ese—

the

com-ander–heorem

This question is answered when the Galois objects are the Galois rings, groups, Lieand birings of Sections 2–6. The fundamental theorems of the corresponding Galoisgroups, Lie rings, and birings theories are then Theorems 2.1, 3.1, 4.2, 6.1, respect

1.1. Summary by sections

In Section 2, the Jacobson–Bourbaki Galois correspondenceF = KR,R = EndF K

between the setFR of subfieldsF of finite codimension of a fieldK and the setR offinite-dimensional endomorphism ringsR of K [8,16,17] generalizes from fieldsK tocommutative ringsA in the Galois rings correspondence Theorem 2.1. It plays a supporole in the Galois groups theory, Galois Lie rings theory, and Galois birings theory wfollow.

In Section 3, the classical Galois correspondence generalizes from fieldsK to commu-tative ringsA in the Galois groups correspondence Theorem 3.1.

In Section 4, the Jacobson differential Galois correspondenceD = DerF K , F = KD

between the setFD of subfieldsF of finite codimension of a fieldK of prime characteristicp such thatKp ⊆ F and the setD of finite-dimensional restricted derivation rings ofK

[8,16] generalizes from fieldsK to commutative ringsA in the Galois Lie rings correspondence Theorem 4.2.

In Section 5, derivation ring forms are introduced and studied. They lead to aconceptual and self-contained proof of Theorem 4.2—as Corollary 5.1.

In Section 6, the Galois rings theory of Section 2 is transformed into a Galoising theory by replacing the endomorphism ringsEndF A by their biring counterpartPresF A = (EndF A,∆,ε).

In Section 7, it is shown how the Galois correspondence theorems for Galoisgroups, Lie rings, and biringsG acting on a fieldK support Galois descent. They leadtheorems which establish passage fromK-modulesV acted on byG to KG -modulesU =V G —this passage being inverse to Galois ascentV = K⊗KG U fromU toV together withthe corresponding action ofG on V . Their generalizations from fieldsK to commutativeringsA are the Galois descent Theorems 7.1, 7.2, 7.4, 7.5.

Finally, in Section 8, four successively easier problems concerned with deteing simple derivation rings are formulated and considered. The hardest of thProblem 8.1—is equivalent to the problem of determining all simple Lie rings. Andeasiest—Problem 8.4—is solved by Theorem 4.2 in the finitely generated case.

1.2. Earlier work

It is instructive to review some of the more closely related earlier work.

1.2.1. Auslander–Goldman Galois extensions. The Chase–Harrison–Rosenbergcorrespondence

Auslander and Goldman introduce and use the notion of Galois extension of amutative ring in [2]. Chase, Harrison, and Rosenberg then adopt and use the AuslGoldman Galois extensions to generalize the classical Galois correspondence t
from fields to commutative rings [4].


eory—bient

Galois

y

bient

arrison–

auto-

rings

Specifically, for a commutative ringS and finite groupG of automorphisms ofS withfixed subringR ≡ SG, S is an Auslander–Goldman Galois extension ofR with GaloisgroupG if S is G-strong. Here, anR-subalgebraT of S is G-strong if for any g,h ∈ G,the restrictions ofg,h to T are equal if and only ifg(t)e = h(t)e for all t ∈ T and allidempotentse of T . The first part of [4, Theorem 2.3] then goes as follows.

Theorem 1.1 (Chase–Harrison–Rosenberg correspondence theorem). LetS be an Auslan-der–Goldman Galois extension ofR with Galois groupG. ThenG′ �→ SG′

is a bijectionfrom the set of subgroups ofG to the set of separableG-strongR-subalgebras ofS.

The resulting Chase–Harrison–Rosenberg Galois theory is a separable Galois thconcerned with separable extensions—which comes into play relative to an amAuslander–Goldman extensionS of R by Galois groupG. TheG is finite, but not alwaysuniquely determined byS.

In contrast, the Galois groups theory of Section 3 of this paper is a separabletheory which comes into play whenB is any commutative ring andH is any group ofautomorphisms ofB such thatH acting on the spectrum of maximal ideals ofB has somefinite orbit—as explained in Example 3.1. So, the groupsG of Theorem 3.1 are usuallinfinite.

1.2.2. The Chase–Sweedler correspondence. Kreimer–TakeuchiJ -Galois extensionsThe Chase–Sweedler Galois theory of [5] comes into play relative to an am

A-objectS for a finite commutative Hopf algebraA over a subringR of S—rather thanrelative to an Auslander–Goldman extensionS of R by Galois groupG. TheS is a com-mutative ring together with an appropriate structure mapα :S → S ⊗ A.

The Chase–Sweedler correspondence [5, Theorem 7.6] generalizes the Chase–HRosenberg correspondence—and is used to reobtain it.

Kreimer–TakeuchiJ -Galois extensionsB of A overR are introduced forJ a Hopf alge-bra over a commutative ringR which is finitely generated and projective asR-module [13].These generalize Chase–Sweedler extensions to non-commutative algebrasB—and lead toa natural Hopf-algebraic definition of normal basis with applications to systems ofmorphisms, derivations, and higher derivations at prime characteristic.

In contrast, the Galois biring theory of Section 6 of this paper enriches the Galoiscorrespondence of Section 2 between the cofinite-dimensional subfieldsF of a commuta-tive ringA and the Galois ringsR of A by endowing theR corresponding to anF with itsbiring structure. This is done within the biringPresA of preservations ofA—with theR

corresponding toF then beingPresF A.

1.2.3. Knus–Ojanguren descent theoryIn their classical workThéorie de la Descente et Algèbres d’Azumaya[12], Knus and

Ojanguren includes theirDescente galoisienneandDescente radicielle de hauteur unforcommutative algebras.

Their Descente galoisienne[12, pp. 44–49] for a commutative algebra extensionS/R

and a finite groupG of automorphisms ofS such thatS is a projectiveR-module of finite


n

oftenis

ra

he-

ebras

be

e

type such thatEndR S hasG as leftS-module basis delivers anSG-form MG of M for anyS- andG-moduleM with descent action

g(sm) = g(s)g(m) (s ∈ S, m ∈ M, g ∈ G).

In contrast, the Galois groups descent theory of this paper comes into play wheG isany finite-irreducible automorphism group of a commutative ringA—e.g., whereA andG

are constructed in the manner of Example 3.1. For such anA andG, and for anyG-descentmoduleV in the sense of Definition 7.2,V G is anAG-form of V .

No counterpart of theDescent galoisienneassumption described above thatG be anS-basis forEndR(S) is necessary in the hypothesis of Theorem 7.2. To the contrary,an irreducible groupG of automorphisms ofA is notA-independent and this conditionnot met. On the other hand, it is a consequence of the irreducibility ofG onA thatEndAG A

is theA-span ofG.Their Descente radicielle de hauteur un[12, pp. 49–53] for a commutative algeb

extensionS/R of characteristicp and a restricted Lie algebraL of derivations ofS suchthatS is a projectiveR-module of finite type such thatEndR S is generated byL as ringandS-module delivers anSL-form ML of M for anyS- and restrictedL-moduleM withdescent action

d(sm) = d(s)m + sd(m) (s ∈ S, m ∈ M, d ∈ L).

In contrast, the Galois Lie rings descent theory of this paper comes into play whenA andD are constructed in the manner of Example 4.1. ThenA is D-simple and everyD-descentmodule forD in the sense of Definition 7.3 has anAD-form V D by Theorem 7.4.

No counterpart of theDescent radicielle de hauteur unassumption thatEndR(S) begenerated as ring andS-module byD described above is needed in the hypothesis of Torem 7.4. Instead, it is a consequence of the irreducibility ofD on A thatEndADA is thering generated byD.

1.3. Conventions

General references are [1,8,16] for fields, [7] for Lie algebras, and [14,16] for coalgand bialgebras over fields.

For a vector spaceV over a fieldF , the zero subspace is 0. The dimension ofV overFis V :F . And the identity endomorphism ofV is IV .

RingsR are assumed to be unital and associative. AndR-modules are assumed tounital—as areR-module homomorphisms.

A subfieldof a ringR is a unital subring ofR which is a field.An algebraover a fieldF is a ringR containingF as central subfield together with th

induced structure as vector space overF .A commutativering (respectively algebra) is a nonzero ring (respectively algebra)A =

(A,π,1A) such thatab = ba (a, b ∈ A).A quasi-local ringis a commutative ring having only finitely many maximal ideals.
Throughout the paper,A denotes a commutative ring.


s

sln

f

For the purposes of this paper, aLie ring is a Lie algebra over the prime fieldπ ≡ Zp

of prime characteristicp or the prime fieldQ of rational numbers—since the Lie ringconsidered in this paper are algebras over some field. Alinear Lie ring is a Lie subalgebraof a linear Lie algebra on a vector spaceV over a prime field.

A restricted linear Lie ringis a linear Lie ringL of prime characteristicp such thatLp ⊆ L.

2. Galois rings theory

The Jacobson–Bourbaki theorem establishes a bijective Galois correspondence

F �→ R ≡ EndF K, R �→ F ≡ KR

between the set of subfieldsF of finite codimension of a fieldK and the set of subringR of the endomorphism ringEndK of K which containKIA and are finite-dimensionaoverK . In its most concise form, the Jacobson–Bourbaki theorem states that such aR isthe algebraEndF K of endomorphisms ofK over the centralizerF = KR of R—and thatF is of finite codimension inK .

The Jacobson–Bourbaki theorem generalizes from fieldsK to commutative ringsA asfollows.

Definition 2.1. An endomorphism ringof A is a subringR of the ring EndA of endo-morphisms ofA which containsAIA. Its centralizeris the subringAR ≡ {b ∈ A | r(ab) =r(a)b (r ∈ R, a ∈ A)} of A.

An endomorphism ringR of a commutative ringA is irreducibleif 0 andA are the onlyR-stable ideals ofA—in which case its centralizerAR is a field by Schur’s lemma.

An endomorphism ringR of A is regarded asA-module—and is said to befinitelygenerated overA if it is finitely generated asA-module.

Definition 2.2. A Galois ring of A is an irreducible endomorphism ring ofA which isfinitely generated overA. A Galois ring subfieldof A is a subfieldF of A of finitecodimension—and the corresponding extension is aGalois ring extension.

Evidently, the map pair(A−,End− A) is a Galois correspondence—sendingF to R =EndF A and R to F = AR—between the set of unital subringsF of A and the set oendomorphism ringsR of A. TheGalois ring correspondence theorem—Theorem 2.1—establishes a bijective Galois correspondence within this one.

Lemma 2.1. Let s1, . . . , sn ∈ EndA. Suppose thatAs1 +· · ·+Asn containsr1, . . . , rm andA containse1, . . . , em such thatriek = δik (1 � i, k � m). Thenm � n.

∑
Proof. Writing ri = j aij sj and applying both sides toek leads to


l

, for

hl

δik = riek =∑j

aij sj ek (1� i, k � m).

This may be expressed as the matrix equation(aij )(sj ek) = I—I being them × m iden-tity matrix overA. SinceA is unital and nonzero,A has a maximal idealM—by Zorn’slemma—and there is a homomorphisma �→ a from A to the fieldA ≡ A/M whose kerneis M . Under this homomorphism, the matrix equation becomes(aij )(sj ek) = I , I beingthem × m identity matrix overA. Since(aij ) is then anm × n matrix of rankm over afield, it follows thatm � n. �2.1. Galois rings correspondence

Theorem 2.1 (Galois rings correspondence theorem). Let R be a Galois ring ofA. ThenF ≡ AR is a subfield ofA of finite codimension andR is the ring EndF A of endomor-phisms ofA overF .

Proof. SinceR is finitely generated overA, R = As1 + · · · + Asn with s1, . . . , sn ∈ R forsome positive integern. SinceR is irreducible and containsAIA, the centralizerEndR A

of R in EndA is F = AR . So, by the Jacobson–Chevalley density theorem [6,9]m � 1 and linearly independente1, . . . , em ∈ A overF , there existr1, . . . , rm ∈ R such thatriek = δik (1� i, k � m). By Lemma 2.1, it follows thatm � n. Taking such a system witm maximal, thee1, . . . , em constitute a basis forA over F—andA is finite-dimensionaoverF . TheEij ≡ eirj (1 � i, j � m) then form the basis forEndF A over F such thatEij ek = δjkei (1 � i, j, k � m). SinceR containsFIA and theEij (1 � i, j � m), it fol-lows thatR = EndF A. �

Theorem 2.1 establishes theGalois rings correspondencefor a commutative ringA—the bijective Galois correspondence

F �→ R ≡ EndF A, R �→ F ≡ AR

between the setFR of Galois ring subfieldsF of A and the setR of Galois ringsR of A.

Example 2.1. Let B be any commutative ring and letS be an endomorphism ring ofBwhich is finitely generated overB. Let J be any maximalS-stable ideal ofB and takeA ≡ B/J . Then the ringR of endomorphisms ofA induced by those ofS onB is a Galoisring of A. So, the hypothesis of Theorem 2.1 is satisfied and the conclusion holds.

3. Galois groups theory

The classical Galois correspondence theorem generalizes from fieldsK to commutative
ringsA as follows.


n

d

ns

Ga-d

-

f

Definition 3.1. An automorphism groupof A is a subgroupG of the groupAut(A) ofautomorphisms ofA. Its centralizeris the subring

AG ≡ {b ∈ A | g(ab) = g(a)b (g ∈ G,a ∈ A)

} = {b ∈ A | g(b) = b (g ∈ R)

}of A.

An automorphism groupG of A is irreducible if 0 andA are the onlyG-stable idealsof A—in which case its centralizerAG is a field by Schur’s lemma.

A subfieldF of A is group irreducibleif the groupAutF A of automorphisms ofAfixing all elements ofF is an irreducible automorphism group ofA.

A Galois group subfieldof A is a group irreducible subfieldF of A such thatF =AAutF A—and the corresponding extensionA/F is a Galois group extension. So, whenA/F is a Galois group extension andA is a field,A/F is simply a Galois field extensioof F .

A Galois groupof A is a group of the formAutF A whereF is a Galois group subfielof A.

By virtue of these definitions, the map pair(A−,Aut− A) is a bijectiveGalois corre-spondenceG = AutF A, F = AG between the setFG of Galois subfieldsF of A and thesetG of Galois groupsG of A.

It remains only to describe the Galois groupsG of A and the corresponding extensioA overAG.

WhenA is quasi-local, the following theorem reduces the problem of describing thelois groupsG of A and corresponding extensionsA overAG—in the broad sense adoptehere—to that of describing them whenA is a field. In fact, most suchA have uncountableGalois groupsG, in which caseA andG are described in terms of the field idealsAi of A

and their automorphism groupsGi—the latter then also being uncountable.

Theorem 3.1 (Galois groups correspondence theorem). Suppose thatA is quasi-local andG is an irreducible automorphism group ofA with fixed fieldF = AG. Then

1. A is the direct sumA = ∑i Ai of finitely many idealsAi which are pairwise isomor

phic Galois field extensions ofF .2. Relative to field isomorphismsαi :Ai → A1 (1 � i � n), AutF A is the internal semi-

direct productP∏

i Gi where P is the symmetric group acting onA by g(a) ≡∑i αg(i)

−1αi(ai) for a = ∑i ai ∈ A = ∑

Ai and∏

i Gi is a normal subgroup oAutF A whereGi acts as the Galois group AutF Ai on Ai and as the identity onAj

(j �= i).3. The subgroupP of AutF A is unique up to conjugacy by the element of

∏i Gi corre-

sponding to another choice of isomorphismsαi .4. P is a Galois group ofA whose corresponding Galois group subfieldF ≡ AP is

isomorphic to thefield idealsAi—andA is isomorphic toFn.

Proof. SinceG is irreducible and stabilizes the intersection⋂

i Mi of the finitely many⋂ ⋂
maximal idealsMi of A, the intersection i Mi is 0. DefiningAi ≡ j �=i Mj , A is then


set

p

s

s

f

d the

the direct sumA = ∑i Ai of the finitely many idealsAi—which are field extensions ofF

by virtue of the direct sum decompositionsA = Ai ⊕ Mi . SinceG is irreducible and thesumB of theAi which are isomorphic toA1 is aG-stable ideal ofA, B = A and theAi

are pairwise isomorphic. The subgroupP of AutF A acts as the symmetric group on theof field ideals{A1, . . . ,An}—with g ∈ P mappingAi to Ag(i) for all i. The product

∏i Gi

of Gi acting asAutF Ai on Ai and as the identity onAj (j �= i) is a normal subgroupof AutF A. For anyx ∈ AutF A,

∏i Gi containsz ≡ y−1x wherey is the element ofP

which permutes theAi the same way as doesx—sincez stabilizes theAi . So,x = yz andAutF A = P

∏i Gi . The stated unicity ofP follows at once. SinceF = AP

∏i Gi , the fields

Ai are Galois extensions ofF with Galois groupsGi .SinceP acts transitively on the field idealsAi , and since any nonzero ideal ofA contains

a field ideal, the onlyP -stable ideal ofA is A. So,P is an irreducible automorphism grouof A. The fieldF = AP is a “diagonal” of the direct sumA = ∑

i Ai which is isomorphic totheAi . To see this, interpreta = ∑

i ai as a linear combinationa = ∑i aiei of basis vectors

ei with coefficients in a field isomorphic to theAi and take the field ideal isomorphismαi :Ai → A1 to beαi(aiei) = aie1 (1 � i � n). Then the action

g

(∑i

ai

)≡

∑i

αg(i)−1αi(ai)

interprets as

g

(∑i

aiei

)≡

∑i

αg(i)−1αi(aiei) =

∑i

αg(i)−1(aie1) =

∑i

aieg(i).

So,

g

(∑i

aiei

)=

∑i

aieg(i) (g ∈ P)

and∑

i aiei ∈ F = AP if and only if the coefficientsai are all equal. This shows thatF isa “diagonal” isomorphic to the field idealsAi andA is isomorphic toFn.

Any automorphism ofA which leaves fixed the elements ofF = AP is of the formx = yz with y ∈ P andz ∈ ∏

i Gi . Sincez = y−1x fixes them as well,z also must fix allelements of allAi , that is,z must be the identity andx = y ∈ P . So,P = AutF A andP isa Galois group ofA. �Example 3.1. Let B be any commutative ring and letH be any group of automorphismof B such thatH acting onSpecmax(B)—the spectrum of maximal ideals ofB—has somefinite orbit Mi . Let J be any maximalH -stable ideal ofB containing the intersection otheMi , takeA ≡ B/J , and takeG to be the group of automorphisms ofA induced byH .ThenA is quasi-local andG-simple. So, the hypothesis of Theorem 3.1 is satisfied an
conclusion holds.


s

-

f

o-

ld

th

-

In particular, whenH is any finite group of automorphisms of any commutative ringB,every orbit ofH in Specmax(B) leads to correspondingA andG satisfying the hypothesiof Theorem 3.1.

Corollary 3.1. Suppose thatG is a finite-irreducible automorphism group ofA and letF ≡ AG. Then

1. A is the direct sumA = ∑i Ai of finitely many idealsAi which are pairwise isomor

phic finite-dimensional Galois field extensions ofF .2. Relative to field isomorphismsαi :Ai → A1 (1 � i � n), AutF A is the internal semi-

direct productP∏

i Gi whereP is a the symmetric group acting onA by g(a) ≡∑i αg(i)

−1αi(ai) for a = ∑i ai ∈ A = ∑

Ai and∏

i Gi is a normal subgroup oAutF A whereGi acts as the Galois group AutF Ai on Ai and as the identity onAj

(j �= i).3. The subgroupP of AutF A is unique up to conjugacy by the element of

∏i Gi corre-

sponding to another choice of isomorphismsαi .4. P is a Galois group ofA whose corresponding Galois group subfieldF ≡ AP is

isomorphic to the field idealsAi—andA is isomorphic toFn.

Proof. The A-spanR = AG of G is an endomorphism ring ofA since (ag)(bh) =ag(b)gh (a, b ∈ A; g,h ∈ G). SinceG is finite and irreducible,R is then a Galois ringof A. Moreover,AR = AG = F by the computations

b ∈ AR ⇒ g(b) = g(1b) = g(1)b = b (g ∈ G) ⇒ b ∈ AG,

b ∈ AG ⇒ g(ab) = g(a)g(b) = g(a)b (a ∈ A, g ∈ G) ⇒ b ∈ AR.

By Theorem 2.1,A : F is finite. So,A has only finitely many maximal ideals and Therem 3.1 applies. �Remark 3.1. When A has only finitely many maximal ideals andG is an irreducibleautomorphism group ofA with finite orbits, theAi in Theorem 3.1 are algebraic fieextensions of the fieldF ≡ AG. To see this, simply note that the polynomialf (X) ≡(X − a1) . . . (X − am) whose roots are the elementsa1, . . . , am of the orbit ofa ∈ Ai underG is fixed byG, so its coefficients are inF . But thena satisfies a nonzero polynomial wicoefficients in the fieldF . So, every elementa of the field extensionAi of F is algebraicoverF .

Remark 3.2. WhenA has only finitely many maximal ideals,G is an irreducible automorphism group ofA, andA is algebraic over the fieldF ≡ AG, then theAi in Theorem 3.1are algebraic field extensions ofAG. ThenG is the product of the finite groupP and anormal subgroup

∏i Gi whose orbits are finite, so the orbits ofG are finite as well.

Remark 3.3. SupposeA is the direct sumA = ∑i Ai of finitely many idealsAi which are

algebraically closed field extensions of a fieldF . Relative to isomorphismsαi :Ai → A1


sr

e

rem

tes that

ve

(1 � i � n), regard the symmetric groupP ≡ Sn as an automorphism group ofA by wayof the actiong(a) ≡ ∑

i αg(i)−1αi(ai) for a = ∑

i ai ∈ A = ∑Ai and g ∈ P . ThenP

is irreducible—since any nonzeroP -stable ideal ofA contains someAi , hence containall Ai , by the transitivity ofP on theAi . So,P is a Galois group ofA—as is the largeAutF A. The corresponding Galois group subfieldF ≡ AP is an algebraic closure ofF—being isomorphic to theAi—with P = AutF A. It depends onP . The purely inseparablclosureFrad of F in F is the Galois group subfieldAAutF A—and so does not depend onP .LettingAsepbe the sum of the separable closures ofF in theAi , A = Asep⊗F Frad (internaltensor product overF ).

4. Galois Lie rings theory

For fieldsK of prime characteristicp, the Jacobson differential correspondence theoestablishes a bijective Galois correspondence

F �→ D ≡ DerF K, D �→ F ≡ KD

between the set of subfieldsF of finite codimension of a fieldK such thatKp ⊆ F andthe set of Lie subringsD of the derivation ringDer(K) of K which containKI , are finite-dimensional overK , and are restricted (as linear Lie algebras over the prime field).

In its most concise form, the Jacobson differential correspondence theorem stasuch aD is the Lie algebraDerF K of derivations ofK over the centralizerF = KD ofD—and thatF is of finite codimension inK .

The Jacobson differential correspondence theorem generalizes from fieldsK to com-mutative ringsA of prime characteristicp as follows.

Definition 4.1. A derivation ringof a commutative ringA is anA-submodule and LiesubringD of the ringDer(A) of derivations ofA. Its centralizeror ring of constantsis thesubring

AD ≡ {b ∈ A | d(ab) = d(a)b (d ∈ D, a,b ∈ A)

} ≡ {b ∈ A | d(b) = 0 (d ∈ D)

}.

A derivation ringis the derivation ring of some commutative ringA.

Evidently, the map pair(A−,Der− A) is a Galois correspondenceD = DerF A,F = AD between the set of unital subringsF of A and the set of derivation ringsD ofA. The Galois Lie ring correspondence theorem—Theorem 4.2—establishes a bijectiGalois correspondence within this one.

Definition 4.2. A derivation ringD of A is irreducible if D �= 0 and 0,A are the onlyD-stable ideals ofA. An irreduciblederivation ring is an irreducible derivation ringD ofsomeA. A finitely generatedirreducible derivation ring is an irreducible derivation ringD

of someA which is finitely generated asA-module.


-ly

-

ce

tohat

l-

The ring of constantsAD of an irreducible derivation ringD of A is a field by Schur’slemma.

4.1. Galois Lie rings and their dual generating systems

Definition 4.3. A Galois Lie ringof A is a restricted irreducible derivation ringD of A

which is finitely generated asA-module. AGalois Lie ring subfieldof A is any subringFof A of the formF = AD whereD is a Galois Lie ring ofA—and the corresponding extensionA/F is called aGalois Lie ring extension. The set of Galois Lie rings (respectiveGalois Lie ring subfields) ofA is denotedD (respectivelyFD).

Definition 4.4. A dual system of rankm in a derivation ringD of A is a paired set of elementsd1, . . . , dm ∈ D, e1, . . . , em ∈ A such thatdiek = δik (1 � i, k � m). A dual systemof A is a dual system in the derivation ringDer(A) of A.

Definition 4.5. A dual generating systemfor a derivation ringD of A is a dual systemd1, . . . , dm ∈ D, e1, . . . , em ∈ A in D such thatD = Ad1 + · · · + Adm.

For D ∈ D, D contains a dual system of rank 0—the vacuous one. Moreover, sinD

is generated asA-module by a finite numbern of elements, every dual system inD hasrankm � n—by Lemma 2.1. From these two facts,D has amaximal dual system—one ofmaximal rank. Consequently, the following theorem establishes, in particular, thatD has adual generating system.

Theorem 4.1. For D ∈ D, every maximal dual system inD is a dual generating systemfor D.

Proof. Suppose thatd1, . . . , dm ∈ D, e1, . . . , em ∈ A with m � 0 is a maximal dual systemin D. Let D′ = {d ′ ∈ D | dej = 0 (1 � j � m)}. ThenD = Ad1 + · · · + Adm + D′. Afterall, eachd ∈ D can be written asd = d(e1)d1 + · · · + d(em)dm + d ′ for somed ′—and thatd ′ vanishes ate1, . . . , em sincedej = (d(e1)d1 + · · · + d(em)dm)ej (1� j � m).

The maximality ofm leads to the conclusionD′ = 0—in two steps. The first step isshow thatD′ mapsA into the nil radicalN of A; and the second step is then to show tA(D′A) is aD-stable ideal ofA contained inN . SinceD is irreducible, the idealA(D′A)

must then be 0, that is,D′ = 0.For the first step, suppose thatD′A is not contained inN and taked ∈ D′, em+1 ∈ A

such thatf ≡ d(em+1) is not nilpotent. Since the derivations inD vanish atpth powers,a ≡ f p is in AD = F . Sincef is not nilpotent,a �= 0. But then 1= a−1f p−1f andf isinvertible inA. Replacingd by dm+1 ≡ f −1d , dm+1em+1 = 1. Sinced ∈ D′, dm+1ej = 0(1 � j � m). But the existence of suchdm+1, em+1 contradicts the maximality ofm. So,D′A is contained inN .

For the second step, note thatD′ is a Lie subalgebra ofD whose Lie algebra normaizer E in D contains thed1, . . . , dm. After all, for d ′ ∈ D′ and 1� i, j � m, [di, d

′]ej =did

′ej − d ′diej = di0 − d ′δij = 0. It follows thatD = Ad1 + · · · + Adm + D′ ⊆ AE ⊆
D and D = AE. SinceE normalizesD′, it stabilizesD′A by the identityd(d ′a) =


al

-

the

ll

d

blylera

l

[d, d ′]a − d ′(da) ∈ D′A (d ∈ E, d ′ ∈ D′, a ∈ A). But then it also stabilizes the ideA(D′A) by the identityd(ab) = d(a)b + ad(b) ∈ A(D′A) + A(E(D′A)) ⊆ A(D′A)

(d ∈ E, a ∈ A,b ∈ D′A). Finally, sinceD′A ⊆ N by the first step, theE-stable idealA(D′A) is contained inN—and so is a properE-stable ideal. SinceD = AE, it is then aproperD-stable ideal as well. SinceD is irreducible, it follows thatA(D′A) is 0—and soD′ = 0 as well.

SinceD = Ad1 + · · · + Adm + D′, the conclusion thatD′ = 0 establishes the decompositionD = Ad1 + · · · + Adm. So,d1, . . . , dm ∈ D, e1, . . . , em ∈ A is a dual generatingsystem forD. �Lemma 4.1. For D ∈ D, A is finite-dimensional overAD .

Proof. The methods of [8, Theorem 19, p. 186] generalize to show that theA-span of themonomialsdi1

1 . . . dimm (0 � i1, . . . , im � p − 1) for d1, . . . , dm anA-module generating se

for D is a Galois ringR. By Theorem 2.1,A is then a finite-dimensional extension of tfield AD . �Definition 4.6. Let B be a commutative ring. Then atruncated polynomial algebraoverB in theei with respect toci ∈ B (1 � i � m) is B[ei | 1 � i � m] ≡ B[Xi | 1 � i � m]/〈Xi

p = ci | 1 � i � m〉. When theci are all 0,B[ei | 1 � i � m] is a truncated polynomiaalgebra overB. When theci are not all 0,B[ei | 1 � i � m] is a truncated polynomiaalgebra overB with respect to constants.

Theorem 4.2 (Galois Lie rings correspondence theorem). For D ∈ D, A is a finite-dimensional truncated polynomial algebra overF ≡ AD with respect to constants—anD = DerF A.

Proof. By Lemma 4.1,A is finite-dimensional overF . Consequently,A has a minimalideal. SinceA is also differentiably simple,A is a truncated polynomial algebraover somesimple ringB—in the sense of Definition 4.6—by the Block’s theorem on differentiasimple algebras with minimal ideal [3]. This simple ringB must be a purely inseparabfield extension ofF—sinceAp ⊆ F . It follows thatA is a truncated polynomial algeboverF with respect to constants. To see thatD = DerF A, suppose thatd1, . . . , dm ∈ D,e1, . . . , em ∈ A with m � 0 is a dual generating system forD. SinceA is finite-dimensionaover F , DerF A is in D. So, it is possible to extend the dual generating system forD toa maximal dual systemd1, . . . , dm′ ∈ DerF A, e1, . . . , em′ ∈ A in DerF A—with m′ � m.The extended system is then a dual generating system forDerF A by Theorem 4.1. Ifm′ >m, thendiem+1 = 0 (1 � i � m) andem+1 ∈ AD = F = ADerF A. But this is impossiblesincedm+1em+1 = 1 is not 0. But thenm′ = m—so thatd1, . . . , dm ∈ D, e1, . . . , em ∈ A isa dual generating system for bothD andDerF A. This establishes thatD = DerF A. �

Theorem 4.2 establishes theGalois Lie ring correspondencefor a commutativering A—the bijective Galois correspondence

F �→ D ≡ DerF A, D �→ F ≡ AD


l

f

s rings withm 5.3nds

ence

between the setFD of Galois Lie ring subfieldsF of A and the setD of Galois Lie ringsDof A—with F being a Galois Lie ring subfield ofA if and only if A is a finite-dimensionatruncated polynomial algebra overF .

Example 4.1. Let B be any commutative ring and letE be a restricted derivation ring oB which is finitely generated overB. Let J be any maximalE-stable ideal ofB and takeA ≡ B/J . Then the derivation ringD of A induced byE is a Galois Lie ring ofA. So, thehypothesis of Theorem 4.2 is satisfied and the conclusion holds.

5. Galois Lie rings theory and derivation ring forms

The Galois Lie ring correspondence Theorem 4.2 was proved using the Galoicorrespondence Theorem 2.1 and Block’s theorem on differentiably simple algebraminimal ideal. It also surfaces independently as Corollary 5.1 to the main Theoreof the following Lie ring Galois theory based on derivation ring forms—which depeneither on Theorem 2.1 nor on Block’s theorem.

Definition 5.1. A derivation ring formof A is a commutative Lie subringL of Der(A)

such thatF ≡ AL is a field,L is finite-dimensional overF , andLp = 0.

An F -formof a moduleD for a commutative algebraA over a fieldF is anF -subspaceL of D such that the bilinear pairing overF sendinga, d (a ∈ A,d ∈ L) to ad presentsD as an internal tensor productA ⊗F L. So, whenL is finite-dimensional overF , L is anF -form of D if and only if any basisd1, . . . , dm for L overF is a free basis forAL overA.

Lemma 5.1. Let L be a derivation ring form ofA with F = AL. ThenL is anF -form ofAL.

Proof. Suppose thata1d1 + · · · + amdm = 0 where theai are inA with not all of them 0.SinceL is abelian, it follows thatx(a1)d1 + · · · + x(am)dm = 0 for all x ∈ L. Since thep-powers of thex ∈ L are 0, there exists a productxn . . . x1 of maximal lengthn � 0of xj ’s in L such that thebi ≡ xn . . . x1ai are not all 0. SinceL is abelian, successivapplication of thex1, . . . , xn to the equationa1d1 + · · · + amdm = 0 leads to the equatiob1d1 + · · · + bmdm = 0 where thebi are inAL = F—contrary to the linear independenof thedi overF . So, thedi form a free basis forD overA. �Theorem 5.1. For D ∈ D with F = AD , D contains a derivation ring formL of A whichis anF -form ofD. In fact,L ≡ Fd1 + · · · + Fdm is such a derivation ring form ofA forany dual generating systemd1, . . . , dm ∈ D, e1, . . . , em ∈ A for D. Conversely, whenL isa derivation ring form ofA, D ≡ AL is in D.

Proof. By Theorem 4.1,D has a dual generating systemd1, . . . , dm ∈ D, e1, . . . , em ∈ A.For any such dual generating system, letL ≡ Fd1 +· · ·+Fdm. ThenL is a derivation ring
form of A. To see this, note that the evident equalities


-

o

ing

e

ces

so

didj (ek) = di(δjk) = 0= dj (δik) = djdi(ek) (1 � i, j, k � m),

dpi ek = d

p−1i (δik) = 0= 0(ek) (1 � i, k � m)

lead to the equalities

didj = djdi (1� i, j � m),

dpi = 0 (1 � i � m)

since theek (1 � k � m) separate the elements ofD. So, L is a commutative finitedimensional Lie subalgebra ofDerF A such thatLp = 0. SinceD = AL, AL = AD = F

andL is a derivation ring form which is anF -form of D by Lemma 5.1.For the converse, suppose thatL is a derivation ring form ofA and setD ≡ AL. Then

D is a derivation ring ofA which is finitely generated asA-module, so it remains only tshow thatD is irreducible onA. SinceD containsL, it suffices to show instead thatL isirreducible onA. To this end, letB be a nonzeroL-stable ideal ofA. SinceL is nonzerofinite-dimensional abelian withLp = 0,L acts as a finite-dimensional space of commutnil linear transformations onB and there exists a nonzerob ∈ B with Lb = 0. Thenb is anonzero element of the fieldAL = F . Consequently,B containsFb = F and 1∈ B. But anidealB containing the identity ofA must beA. So,B = A andD is irreducible onA. �Theorem 5.2. Let d ∈ Der(A) and e ∈ A wheredp = 0 and d(e) = 1—and letB ≡ Ad .ThenA = B[e] andA is the truncated polynomial algebra ine overB with respect to somc ∈ B, that is, there is an isomorphism overB fromA to B[X] ≡ B[X]/〈Xp = c〉 mappinge to X ≡ X + 〈Xp = c〉.

Proof. Sinced(ep) = 0, c ≡ ep is in Ad = B. There can be noB-linear relationb0 +b1e + · · · + bne

n = 0 of the powers 1, e, . . . , ep−1 of e with some nonzerobi and minimallengthn � p − 1—since there would then be one with shorter length, namely,b1 + 2b2e +· · · + nbne

n−1 = 0 obtained by applyingd . So, the powers 1, e, . . . , ep−1 of e are linearlyindependent overB andB[e] is the truncated polynomial algebra ine overB with respectto c.

To show thatA = B[e], it suffices to show by induction that the Engel subspaA0 ≡ F , An+1 ≡ {a ∈ A | d(a) ∈ An} of the nilpotent derivationd on A with dp = 0 arecontained in those forB[e]—namelyB[e]0 ≡ F , B[e]n+1 ≡ {f ∈ B[e] | d(f ) ∈ B[e]n}—for 1 � n � p − 2. Forj = 0, A0 = F = B[e]0 by definition—so assume 0< j � p − 2andAj = B[e]j . Take anya ∈ Aj+1. Thenda ∈ Aj = B[e]j . Integratingd(a) in B[e]jwith respect tod producesf ∈ B[e]j+1 such thatd(a) = d(f ). But thend(a − f ) = 0and a − f ∈ Ad = B. So, thea ∈ f + B are contained inB[e]j+1. This being truefor all sucha, it follows that Aj+1 ⊆ B[e]j+1. This completes the induction step,
A = Ap−1 = B[e]p−1 = B[e]. �


,-

as

d

t

Theorem 5.3. LetL be a derivation ring form ofA withF = AL. Then DerF A ∈ D, L is anF -form of DerF A, and for every basisd1, . . . , dm for L overF , there existe1, . . . , em ∈ A

such thatd1, . . . , dm ∈ L, e1, . . . , em is a dual system ofA with

A = F [e1] ⊗F · · · ⊗F F [em],

andF [ei] is the truncated polynomial algebra inei with respect to someci ∈ F for 1 �i � m.

Proof. The proof is by induction onm—and is evident whenm = 0 with A = F . Ifm = 1, thenL = Fd1, F = Ad1, there existse1 such thatd1(e1) = 1, andA = F [e1]is the truncated polynomial algebra with respect to somec1 ∈ F by Theorem 5.2. ThusDerF A = F [e1]d1 = AL andDerF A ∈ D. Suppose next thatm > 1—and that the theorem holds for derivation ring forms of algebras of dimensionm − 1 or less. For 1� i � m,

let Li ≡ Fd1 +· · ·+︷︸︸︷Fdi +· · ·+Fdm (sum ofm terms with theith removed). IfALi = F ,

then DerF A = ALi by the induction hypothesis. But thendi ∈ ALi . This contradictsLemma 5.1, according to which theF -basisd1, . . . , dm for L is aA-basis forAL. It fol-

lows from this and Theorem 5.2 thatALi �= F with ALidi = F andALi = F [ei] for some

ei ∈ ALi with di(ei) = 1—for 1� i � m. But then the dualitydj (ei) = δij holds and

(1) A = Adm [em]—truncated polynomial algebra inem overAdm with respect tocm = epm;

(2) Adm = F [e1]⊗F · · ·⊗F F [em−1]—where theF [ej ] are truncated polynomial algebrin ej overF with respect tocj = e

pj .

This is by Theorem 5.2 and the induction assumption. Consequently,DerF A ∈ D, L is anF -form of DerF A, and there existe1, . . . , em ∈ A such thatd1, . . . , dm ∈ L, e1, . . . , em isa dual system ofA with

A = Adm [em] = F [e1] ⊗F · · · ⊗F F [em−1][em] = F [e1] ⊗F · · · ⊗F F [em],

where theF [ei] are truncated polynomial algebras inei with respect toci ∈ F for any1� i � m. �

Theorem 4.2 now reappears as the following corollary.

Corollary 5.1 (Galois Lie rings correspondence theorem). For D ∈ D, A is a finite-dimensional truncated polynomial algebra overF ≡ AD with respect to constants—anD = DerF A.

Proof. Every Galois Lie ringD of A has a derivation ring formL by Theorem 5.1. Bu
thenD = DerF A andA is a truncated polynomial algebra overF by Theorem 5.3. �


gs

f

6. Galois birings theory

The Galois rings correspondence

F �→ R ≡ EndF A, R �→ F ≡ AR,

between the finite-codimensional subfieldsF of a commutative ringA and the Galois ringsR of A can be enriched by imposing biring structures on the Galois ringsR of A whichreflect the structures of the corresponding ring extensionsA/AR .

6.1. The biring of preservations ofA overF

The preservation sets ofA over a subfieldF are the counterparts for commutative rinof the coclosed sets ofK/k-bialgebras of [15,16].

Definition 6.1. A preservation setof A over F is a subsetH of EndF A such that foreachx ∈ H there exist finitely manyix, xi in H such thatx(ab) = ∑

i ix(a)xi(b) for alla, b ∈ A.

When H and H ′ are preservation sets ofA over F , so is the setHH ′ of productshh′ (h ∈ H, h′ ∈ H ′). For if finitely manyx andix, xi in H satisfyx(ab) = ∑

i ix(a)xi(b)

for all a, b ∈ A, and finitely manyy andj y, yj in H ′ satisfyy(ab) = ∑j j y(a)yj (b) for

all a, b ∈ A, then the finitely manyxy andixj y, xiyj in HH ′ satisfy

(xy)(ab) =∑i,j

ixj y(a)xiyj (b)

for all a, b ∈ A.Arguing in like fashion, whenH andH ′ are preservation sets ofA overF , so are the

set−H ′ of negatives−h′ (h′ ∈ H ′) and the setH + H ′ of sumsh + h′ (h ∈ H,h′ ∈ H ′).It follows that PresF A is a subring ofEndF A. Moreover, since the sets{aIA, IA}

(a ∈ A) are preservation sets ofA overF , it containsAIA. So,PresA is an endomorphismring of A in the sense of Definition 2.1.

As the union of preservation sets ofA over F , PresF A, too, is a preservation set oA overF . The following lemma shows, forF finite-codimensional, thatPresF A may beendowed with the structure of coalgebra overA.

Lemma 6.1. Suppose thatF is a finite-codimensional subfield ofA. Then finitely manyix, xi and finitely manyj y, yj in EndF A satisfy the equations

∑i

ix(a)xi(b) =∑j

j y(a)yj (b) (a, b ∈ A)

if and only if


∑i

ix ⊗A xi =∑j

j y ⊗A yj .

Proof. Suppose first that∑

i ix(a)xi(b) = ∑j j y(a)yj (b) (a, b ∈ A). Select a basiser

for A over F—and letzr ∈ EndA be defined by the conditionszr(es) = δrs. Then eachz ∈ EndA decomposes uniquely asz = ∑

r z(er )zr—and thezr form a basis forEndA

overA. Writing xi = ∑r xirzr andyj = ∑

r yjrzr for all i andj leads to

∑r

∑i

ix(a)xirzr =∑

r

∑j

j y(a)yjrzr .

By theA-independence of thezr , this, in turn, leads to

∑i

ix(a)xir =∑j

j y(a)yjr (a ∈ A),

∑i

xir ix =∑j

yjr j y

for all r . But then

∑r

(∑i

xir ix

)⊗A zr =

∑r

(∑j

yjr j y

)⊗A zr

and

∑i

ix ⊗A

(∑r

xirzr

)=

∑j

j y ⊗A

(∑r

yjrzr

),

which establishes that∑

i ix ⊗A xi = ∑j j y ⊗A yj .

Suppose, conversely, that∑

i ix ⊗A xi = ∑j j y ⊗A yj . For anya, b ∈ A, the map

(x, y) �→ x(a)y(b)

is anA-bilinear pairing fromEndA × EndA to A. Consequently, there is anA-linear map

ρ : EndA ⊗A EndA → A

such thatρ(x ⊗A y) = x(a)y(b). Since∑

i ix ⊗A xi = ∑j j y ⊗A yj , it follows that

∑i

ix(a)xi(b) =∑j

j y(a)yj (b) (a, b ∈ A). �

ForF a finite-codimensional subfield ofA, Lemma 6.1 ensure that∆(x) ∈ PresF A⊗A

PresF A is well defined forx ∈ PresF A by


coal-

f

e-

at thed

∆(x) ≡∑

i

ix ⊗A xi,

whereix, xi are finitely many elements ofPresF A such that

x(ab) =∑

i

ix(a)xi(b) (a, b ∈ A).

Defining ε(x) ≡ x(1A) (x ∈ PresF A), PresF A = (PresF A,∆,ε) is then acoalgebraoverA—meaning that the coproduct and coidentity maps

∆(x) =∑

i

ix ⊗A xi (x ∈ PresA),

ε(x) = x(1A) (x ∈ PresA)

satisfy the coassociativity and coidentity laws which generalize naturally those forgebras overA whenA is a field [14,16]. These laws forPresF A follow at once from theassociativity and identity laws forA and the above definition of∆ andε.

Definition 6.2. For F a finite-codimensional subfield ofA, thebiring of preservations oA overF is PresF A as ring and coalgebra(PresF A,π, IA,∆, ε) overA.

More generally, wheneverF is a subfield ofA andPresF A may be regarded as coalgbra overA with coidentityε(x) = x(1A) and coproduct∆(x) (x ∈ PresF A) such that

∆(x) ≡∑

i

ix ⊗A xi

if and only if ix, xi are finitely many elements ofPresF A such that

x(ab) =∑

i

ix(a)xi(b) (a, b ∈ A),

(PresF A,π, IA,∆, ε) is called thebiring of preservations ofA over F . Examination ofthe proof of Lemma 6.1 shows that such a∆ exists whenEndF A has anA-basis. So, thiscondition is always met whenEndF A has anA-basis, e.g., whenA is any field extensionof a fieldF .

6.2. Galois birings correspondence

Theorem 2.1 now takes on the form of the following Theorem 6.1, which shows thGalois ringR corresponding to a subfieldF of A of finite codimension may be endowe
with the biring structure(R,∆, ε) = (PresF A,∆,ε).


-

6,

-

Theorem 6.1 (Galois birings correspondence theorem). LetR be a Galois ring of a commutative ringA. ThenF ≡ AR is a subfield ofA of finite codimension andR = PresF A =EndF A.

Proof. By Theorem 2.1,F ≡ AR is a subfield ofA of finite codimension andR = EndF A.So, it remains to show thatPresF A = EndF A. This is seen by revisiting the proof of [1Theorem 5.3.10] in the present context—withA in place ofK . AlthoughF is a field andthe extensionA/F is finite-dimensional, due care must be exercised sinceA is no longerassumed to be a field.

With R = EndF A, let R∗ ≡ R∗(A) ≡ HomA(R,A) be thedual ofR with coefficientsin A. Let theei be a basis forA overF and let theri be the elements ofR defined by thecondition thatri(ej ) = δij . Since eachx ∈ R can be written uniquely asx = ∑

i x(ei)ri ,the ri are anA-basis forR. Defining a ∈ R∗ by a(x) ≡ x(a) (x ∈ R) for a ∈ A, the ei

satisfy ej (ri) = ri(ej ) = δij . So, theei are a dual basis overA for the dualR∗ of R withcoefficients inA. Since theei are also anF -basis for theF -subspaceA ≡ {a | a ∈ A}of R∗, A is anF -form of theR-moduleR∗ andR∗ = AA = A ⊗F A. EndowingR∗ withtheA-algebra product induced by theF -algebra product ofA by ascent fromF to A, theidentity andA-algebra product ofR∗ are given by

1C∗ = 1A, ab = ab (a, b ∈ A).

Since theA-algebraR∗ has a finiteA-basis, it induces a dualA-coalgebra structure onR—with coidentityε defined by

ε(x) ≡ x(1A) (x ∈ R)

and coproduct∆(x) (x ∈ R) defined by the condition

∆(x) ≡∑

i

ix ⊗A xi ⇔ (ab)(x) =∑

i

a(ix) ⊗A b(xi) (a, b ∈ A)

⇔ (ab)(x) =∑

i

a(ix) ⊗A b(xi) (a, b ∈ A)

⇔ x(ab) =∑

i

ix(a) ⊗A xi(b) (a, b ∈ A).

This establishes thatR = EndF A is a preservation set—and, therefore, thatEndF A =PresF A. �Definition 6.3. A Galois biringof a commutative ringA is a Galois ringR of A regardedas the biringR = (PresAR A,π, IA,∆, ε) of preservations ofA overF according to Theo
rem 6.1 and Definition 6.2.


rings)

Galoist

For a coalgebraC = (C,∆C, εC) overA, thedual algebraC∗(A) ≡ HomA(C,A) withcoefficients inA of C is theA-moduleC∗ ≡ C∗(A) together with identity 1C∗ ≡ εC andalgebra productfg = π(f,g) defined by

(fg)(x) ≡∑

i

f (ix)g(xi)

(x ∈ C, ∆C(x) =

∑i

ix ⊗ xi

)

for f,g ∈ C∗. The following corollary is evident from the proof of Theorem 6.1.

Corollary 6.1. For a finite-dimensional commutative algebraA/F , the dual algebraP ∗(A) = HomA(P,A) with coefficients inA of the coalgebraP ≡ PresF A over A isthe extensionAA = A ⊗F A of theF -algebraA to A-algebra by ascent.

Theorem 6.1 establishes theGalois birings correspondencefor a commutative ringA—the bijective Galois correspondence

F �→ R ≡ EndF A, R �→ F ≡ AR

between the subfields ofA of finite codimension and the Galois birings ofA.

7. Galois descent

The Galois correspondence theorems for Galois objects (Galois groups, rings, LieG acting on a fieldK lead toGalois descent theorems—theorems which providemecha-nisms of Galois descentfrom K-modulesV acted on byG to KG -modulesU = V G

[10,16,17]. This descent is inverse toGalois ascentV = K ⊗KG U from U to V togetherwith the corresponding action of the Galois objectG onV .

Now, more generally, the Galois correspondence theorems for Galois objects (groups, rings, Lie rings, birings)G acting on a commutative ringA lead to Galois descentheorems providing mechanisms of Galois descent fromA-modulesV acted on byG toAG -modulesU = V G . Again, this descent is inverse to Galois ascentV = A⊗AG U fromU to V together with the corresponding action of the Galois objectG onV .

7.1. Galois rings descent

Let R be a Galois ring ofA with centralizerF ≡ AR . ThenGalois ring ascentby R

is the passage from anF -spaceU to theA-moduleV ≡ A ⊗F U regarded asR-modulewith respect to the action ofR on V well defined by the conditionr(b ⊗ u) ≡ r(b) ⊗ u

(r ∈ R,b ∈ A,u ∈ U). The resultingV is anR-descent module withR-centralizerV R =1⊗F U in the following sense.

Definition 7.1. For a Galois ringR of A, an R-descent module—or Galois rings de-
scent module forR—is an A-module and (ring)R-moduleV such that(ar)v = a(rv)


m

fieldsm

is

t

(r ∈ R, a ∈ A, v ∈ V ). The centralizerof R in such aV is theAR-submoduleV R ≡{u ∈ V | r(au) = r(a)u (r ∈ R, a ∈ A)} of R in V .

The simplest instance of this is the Galois descent moduleV = A—whoseA-moduleoperation is the multiplication ofA and whoseR-module operation is the endomorphisring action ofR onA.

The Jacobson endomorphism ring descent theorem [10,16,17] generalizes fromto commutative rings as follows—whereA andR might come from any endomorphisring S of a commutative algebraB which is finitely generated overB—as explained inExample 2.1.

Theorem 7.1 (Galois rings descent theorem). LetV be anR-descent module for a Galoring R of A. ThenV R is aAR-form ofV .

Proof. Let F be the fieldAR . By the Galois ring correspondence theorem,R = EndF A.Take a basise1, . . . , en for A over F . Then a = ∑

j rj (a)ej (a ∈ A) where r1, . . . , rnare theF -linear transformations fromA to F such thatrj (ek) = δjk (1 � i, j � n)—the corresponding dual basis. Since the linear transformationsEij ≡ eirj mapek to δjkei

(1� i, j, k � n) they form a basis forR overF . And ther1, . . . , rn form a basis forR overA—with the unique linear combination deliveringr ∈ R beingr = ∑

j r(ej )rj . Moreover,I = ∑

j Ejj = ∑j ej rj . Sincerj mapsA into F , thenr(arj ) = r(a)rj (r ∈ R,a ∈ A,

1� j � n). But then

r(a(rj v)

) = r((arj )v

) = (r(arj )

)v = (

r(a)rj)v = r(a)(rj v),

r(a(rj v)

) = r(a)(rj v)

for r ∈ R, a ∈ A, v ∈ V, 1 � j � n. So, therj map thev ∈ V into V R , v = Iv =∑j ej rj v is in AV R , andV is theA-span ofV R . To show thatV R is a F -form of V ,

it remains only to show thatF -independent elementsvk of V R areA-independent. Buan A-relation

∑k fkvk = 0 leads to theF -relations

∑k(rj fk)vk = 0—since therj map

the fk into F . So, therjfk are all 0. But then thefk = Ifk = ∑j ej rj fk are all 0. This

establishes that thevk areA-independent andV R is anF -form of V . �Example 7.1. Let B be a ring containing a commutative ringA as unital subring. LetRbe a unital subring and finitely generatedA-submodule ofV ≡ EndB which stabilizes andacts faithfully and irreducibly onA. ThenR can be regarded as a Galois ring ofA—andV as anR-descent module withR-actionrv ∈ V (r ∈ R, v ∈ V ). ThenV R is anRF -formof V by Theorem 7.1.

Example 7.2. As an instance of Example 7.1, for a Galois ringR of A and F ≡ AR ,V ≡ EndF A is anR-descent submodule ofEndA. Its centralizer is the dual space

( )
V R = HomF (A,A)R = HomF A,AR = HomF (A,F ) = A∗


lds to

-

of A over F . In this instance, Theorem 7.1 simply reconfirms thatA∗ is anF -form ofEndF A.

7.2. Galois groups descent

Let G be a Galois group ofA with centralizerF ≡ AG. ThenGalois group ascentbyG is the passage from anF -spaceU to theA-moduleV ≡ A⊗F U regarded asG-modulewith respect to the action ofG on V well defined by the conditiong(b ⊗ u) ≡ g(b) ⊗ u

(g ∈ G,b ∈ A,u ∈ U). The resultingV is aG-descent module withG-centralizerV G =1⊗F U in the following sense.

Definition 7.2. For G a Galois group ofA, a G-descent module—or Galois groups de-scent modulefor G on A—is anA-module and (group)G-moduleV such that(ag)v =a(gv), g(av) = g(a)g(v) (g ∈ G, a ∈ A, v ∈ V ) and

∑g∈G agg = 0 ⇒ ∑

g∈G aggλ = 0(g ∈ G,ag ∈ A) wheregλ is the linear transformation ofV defined bygλ(v) ≡ gv (g ∈ V,

v ∈ V ). Thecentralizerof G in such aV is theAG-submoduleV G ≡ {u ∈ V | g(u) = u

(g ∈ G)} of G in V .

The simplest instance of this is the Galois groups descent moduleV = A for G onA—whoseA-module operation is the multiplication ofA and whoseG-module operation isthe automorphism group action ofG onA.

Remark 7.1. The condition

∑g∈G

agg = 0 ⇒∑g∈G

aggλ = 0 (g ∈ G,ag ∈ A)

is thenondegeneracy condition. It is satisfied automatically whenA is a field—in whichcase

∑g∈G

agg = 0 ⇒ ag = 0 (g ∈ G, ag ∈ A)

by Dedekind’s lemma on the linear independence of theg ∈ G overA. WhenA is not afield, theg ∈ G are usually not linearly independent.

The descent theorem of A. Speiser [16, Theorem 3.2.5] generalizes from fiecommutative rings as follows—whereA andG might come from any finite groupH ofautomorphisms of any commutative algebraB as described in Example 3.1.

Theorem 7.2 (Galois groups descent theorem). LetV be aG-descent module for a finiteirreducible automorphism groupG of A. ThenV G is anAG-form ofV .

Proof. Let R be the span ofG over A. ThenR is a Galois ring ofA sinceG is finite-
irreducible and(ag)(bh) = ag(b)gh (a, b ∈ A; g,h ∈ G). The nondegeneracy condition


1,

:

ms4.

,

∑g∈G agg = 0 ⇒ ∑

g∈G aggλ = 0 (g ∈ G,ag ∈ A) ensures that the action ofG on V ex-tends to an action ofR onV relative to whichV is anR-descent module. By Theorem 7.V R is then aV R-form of V . This completes the proof, sinceAG = AR andV G = V R—asshown by the following computations:

u ∈ V R ⇒ g(u) = g(1u) = g(1)u = u (g ∈ G) ⇒ u ∈ V G,

u ∈ V G ⇒ g(au) = g(a)g(u) = g(a)u (a ∈ A, g ∈ G) ⇒ u ∈ V R. �Example 7.3. For G a finite-irreducible automorphism group ofA, the A-moduleV ≡EndA is aG-descent module withG-module action

g(v)(b) ≡ gvg−1b (a, b ∈ A, g ∈ G, v ∈ V ).

In fact, the descent conditiong(av) = g(a)g(v) (a ∈ A, v ∈ V ) is established as follows

g(av)(b) = g(av)g−1b = g(a(v(g−1b)

)) = g(a)g(v(g−1b)

) = g(a)gvg−1b

= (g(a)g(v)

)(b).

Its centralizer

V G = (EndA)G = {v ∈ EndA | g(v) = v (g ∈ G)

}= {

v ∈ EndA | gvg−1 = v (g ∈ G)}

= {v ∈ EndA | gv = vg (g ∈ G)

}is anAG-form of EndA by Theorem 7.2.

Example 7.4. As an instance of Example 7.3, suppose thatA = Fn is the direct sum ofncopies ofF andP is the symmetric group onn letters acting as a group of automorphisof A over the diagonalF and permuting theF factors in the manner of Corollary 3.1 partThen(EndF A)P is anF -form of EndF A.

Example 7.5. ForG a finite-irreducible automorphism group ofA andF ≡ AG, DerF A isaG-descent submodule of theG-descent moduleEndA of Example 7.3. By Theorem 7.2its centralizer(DerF A)G is anF -form of DerF A.

7.3. Galois Lie rings descent

Let D be a Galois Lie ring ofA with centralizerF ≡ AD . Then Galois Lie ringascentby D is the passage from anF -spaceU to the A-module V ≡ A ⊗F U re-garded asD-module with respect to the action ofD on V well defined by the conditiond(b ⊗ u) ≡ d(b) ⊗ u (d ∈ D,b ∈ A,u ∈ U). The resultingV is aD-descent module with
D-centralizerV D = 1⊗F U in the following sense.


n

e

m

,

o

Definition 7.3. A D-descent module—or Galois Lie rings descent modulefor D on A—is anA-module and (Lie ring)D-moduleV such that(ad)v = a(dv), d(av) = d(a)v +a(dv), and(dλ)

p = dpλ (d ∈ D, a ∈ A, v ∈ V )—wheredλ is the linear transformatio

dλ(v) ≡ dv (v ∈ V ) of V . Thecentralizerof D in such aV is theAD-submodule

V D ≡ {u ∈ V | d(au) = d(a)u (d ∈ D, a ∈ A)

} = {u ∈ V | d(u) = 0 (d ∈ D)

}of D in V .

The condition(dλ)p = d

pλ (d ∈ D, a ∈ A, v ∈ V ) is simply the condition that the Li

ring D-moduleV be restricted.The simplest instance of this is the Galois Lie rings descent moduleV = A for D on

A—whoseA-module operation is the multiplication ofA and whoseD-module operationis the derivation ring action ofD onA.

Definition 7.4. A toral form of a Galois Lie ringD is a commutativeAD-Lie subalgebraof D such thatTπ ≡ {t ∈ T | tp = t} is a π -form of D whereπ ≡ {0, . . . , p − 1} is theprime field ofF .

TheTπ of a toral formT of a Galois Lie ringD is aπ -form Tπ of T called theprimeform of T . SinceT is abelian with restricted prime form,T itself is restricted.

Theorem 7.3. Let D be a Galois Lie ringD of A. ThenD has a dual generating systed1, . . . , dm ∈ D, e1, . . . , em ∈ A such thatT ≡ Fe1d1 + · · ·+ Femdm is a toral form ofD.

Proof. Let F be the fieldAD and take a dual generating systemd1, . . . , dm ∈ D,e1, . . . , em ∈ A for D. If ei is nilpotent, sinceep

i is in F , it is 0—so that(ei + 1)p =0+ 1= 1. Since derivations vanish at 1, replacing the nilpotentei by ei + 1 then results ina dual generating systemd1, . . . , dm ∈ D, e1, . . . , em ∈ A for D where theep

i are nonzero

elements ofF . Theei are then invertible with inversee−1i = e

p−1i (e

pi )−1. By Theorem 5.1

L ≡ Fd1 + · · · + Fdm is a derivation ring form ofA and F -form of D. Let ti ≡ eidi

(1 � i � m) andT ≡ F t1 + · · · + F tm. Since theei are invertible,D = AL = AT = ATπ .So,T is also anF -form of D—andTπ ≡ πt1 + · · · + πtm is aπ -form of D. Furthermore,T andTπ are restricted abelian Lie subrings ofD. In fact, the evident equalities

ti tj (ek) = ti (δjkek) = δik(δjkek) = δjk(δikek) = tj (δikek) = tj ti(ek),

tpi (ek) = ti (ek)

for 1� i, j, k � m lead to the equalities

ti tj = tj ti , tpi = ti

for 1� i, j � m since theek (1� k � m) separate the elements ofD. These identities alsshow thatTπ = {t ∈ T | tp = t}—since the coefficientsai ∈ F of anya1t1+· · ·+amtm ∈ T

such that


mmu-

5.2.9]fd

a

l

ap

1 t1 + · · · + apmtm = (a1t1 + · · · + amtm)p = a1t1 + · · · + amtm

satisfy the conditionapi = ai and so are in the splitting fieldπ of the polynomialXp − X.

So,T is a toral form ofD. �The Jacobson differential descent theorem [10,16] generalizes from fields to co

tative rings as the following Theorem 7.4—whereA andD might come from any finitelygenerated derivation ringE of any commutative algebraB as described in Example 4.1.

The proof of Theorem 7.4, motivated by that of the corresponding [16, Theoremfor fields, illustrates how toral formsT of Galois Lie ringsD are used. The irreducibility oD onA makes up for the absence of inverses when the fieldsK of [16] are now generalizeto commutative ringsA.

Theorem 7.4 (Galois Lie rings descent theorem). Let V be a D-descent module forGalois Lie ringD of A. ThenV D is anAD-form ofV .

Proof. Let F = AD , let T be a toral form ofD, and letTπ = πt1 + · · · + πtm be its primeform—theti being a basis forTπ overπ . Sincetp = t for t ∈ Tπ , and sinceV is a restrictedLie module forT , t

pλ = tλ and the separable polynomialXp − X vanishes on thetλ ∈ Tπλ

acting on theπ -spaceV . So, the eigenvalues of thetλ ∈ Tπλ are in the splitting fieldπ ofXp −X and thetλ ∈ Tπλ act diagonalizably onV . SinceTπ is abelian,V then has spectradecompositionV = ∑

σ∈S Vσ whereS is the set ofσ in the π -dual spaceT ∗π of Tπ for

whichVσ ≡ {v ∈ V | tv = σ(t)v} is nonzero.Applying this to the special case where the Galois descent module isA, A too has

spectral decompositionA = ∑α∈R Aα whereR is the set ofα in theπ -dual spaceT ∗

π ofTπ for whichAα ≡ {x ∈ A | tx = α(t)x} is nonzero.

The first part of the proof is to show that the spectral decompositionsA = ∑α∈R Aα

andV = ∑σ∈S Vσ areA = ∑

α∈R xαA0 = ∑α∈R Fxα andV = ∑

α∈R xαV0.Taking nonzeroxα ∈ Aα (α ∈ R), xαA = A for α ∈ R. To see this, note thatxαA is

stable under allt ∈ Tπ by the computation

t (xαb) = t (xα)b + xαt (b) = α(t)xαb + xαt (b) ∈ xαA

for b ∈ A. But thenxαA is also stable underD—sinceD = ATπ and

(at)(xαA) = a(t (xαA)

) ⊆ a(xαA) ⊆ xαA

for a ∈ A, t ∈ Tπ . SinceD is irreducible onA, it follows thatxαA = A (α ∈ R).From A = ∑

α∈R Aα , AαAβ ⊆ Aα+β (α,β ∈ R), andxαA = A (α ∈ R) come the in-clusions

∑Aγ = A = xαA =

∑xαAβ ⊆

∑Aα+β.

γ∈R β∈R β∈R


,of

hat

t

ng

s

Evidently, then, thexαAβ are nonzero andxαAβ = Aα+β for all α,β ∈ R. ConsequentlyAα+β is nonzero andα + β ∈ R for all α,β ∈ R. SinceR is an additively closed subsetthe finite additive groupT ∗

π , R is an additive subgroup ofT ∗π . Sinceπ is the prime field

of p elements, the subgroupR of theπ -spaceT ∗π is aπ -subspace ofT ∗

π . Finally, sinceRseparates the points ofTπ , theπ -subspaceR of T ∗

π is R = T ∗π .

Takingβ = 0 in xαAβ = Aα+β gives

Aα = Aα+0 = xαA0 = xαF = Fxα.

So,Aα = Fxα for α ∈ R andA = ∑α∈R Fxα .

SincexαA = A, xαV ⊇ xαAV = AV = V andxαV = V (α ∈ R)—from which comethe inclusions ∑

γ∈S

Vγ = V = xαV =∑σ∈S

xαVσ ⊆∑σ∈S

Vα+σ .

Evidently, then, thexαVσ are nonzero andxαVσ = Vα+σ for all α ∈ R, σ ∈ S. For σ ∈ S,then,Vα+σ is nonzero andα + σ ∈ S for all α ∈ R. SinceR = T ∗

π , it follows thatR + σ ⊆S ⊆ R. SinceR + σ andR are finite with the same number of elements, it follows tR + σ = S = R, that is,S = R = T ∗

π .The point of all this is that, sinceS = T ∗

π , S contains 0—and then, upon takingσ = 0 inthe equationxαVσ = Vα+σ , that

Vα = Vα+0 = xαV0.

This establishes thatVα = xαV0 for α ∈ R andV = ∑α∈R xαV0—completing the first par

of the proof.Proving thatV D = V0 is anF -form of V reduces to showing that the bilinear pairi

a,u �→ au (a ∈ A, u ∈ V0) from A,V0 to V is a tensor productV = A ⊗F V0. By the firstpart of the proof, the span of the image of this pairing is

AV0 =( ∑

α∈R

Fxα

)V0 =

∑α∈R

xαV0 =∑α∈R

Vα = V.

So, it remains only to show thatA andV0 arelinearly-disjointoverF , that is,F -linearlyindependent elements ofV0 areA-linearly independent.

Suppose, to the contrary, thatA and V0 are not linearly-disjoint overF and choosea set of linearly independentu1, . . . , un with n minimal for which there exist relationa1u1 + · · ·+ anun = 0 overA (with theai being fromA) with not all 0. By the minimalityof n, all theseai are nonzero—includinga1. Consequently, the setJ of all a1 ∈ A forwhich there is a relationa1u1 + · · · + anun = 0 over A is then a nonzero ideal ofA.Since thed ∈ D satisfyd(u) = 0 (u ∈ V0), a relationa1u1 + · · · + anun = 0 overA leadsto corresponding relationsd(a1)u1 + · · · + d(an)un = 0 (d ∈ D) overA—from which itfollows thatJ is a nonzeroD-stable ideal ofA. SinceD is irreducible onA, J = A. So,
J contains 1 and there exists a relationa1u1 +· · ·+anun = 0 overA with a1 = 1. But then


ter

to

ie

escent

ion 6truc-

scent

n be

scentpent

d(a1) = d(1) = 0. So, the relationa1u1+· · ·+anun = 0 leads to the corresponding shorrelationsd(a2)u2 + · · · + d(an)un = 0 (d ∈ D). By the minimality ofn, the coefficientsd(ai) (d ∈ D) are then all 0—anda1, . . . , an ∈ AD = F . Since theai are not all zero, thiscontradicts the linear independence of theui over F . Retreating from the suppositionthe contrary,A andV0 then are linearly-disjoint overF . �Example 7.6. Let V be a ring containing a commutative ringA as unital subring—andregardV asA-module. LetD be anA-submodule and restricted Lie subalgebra ofDerA V

which stabilizes and acts irreducibly onA with the fieldF ≡ AD finite-codimensional—and suppose that restriction ofD to A is faithful. ThenD may be regarded as a Galois Lring of A—andV as aD-descent module. So, its centralizer

V D = {v ∈ V | x(av) = x(a)v (a ∈ A, x ∈ D)

} = {v ∈ V | x(v) = 0 (x ∈ D)

}is anF -form of V by Theorem 7.4.

7.4. Galois birings descent

The Galois birings descent theorem is an important footnote to the Galois rings dtheorem—for the following reasons:

1. Moving from Galois rings theory of Section 2 to the Galois birings theory of Sectamounted largely to endowing Galois rings with biring structures—so that the sture of the ring extensionA/F and the structure of the corresponding biringPresF A

faithfully reflect each other.2. Galois biring ascent changes in name only. And, accordingly, Galois birings de

changes in name only.3. Importantly, however, the nature of the underlying action for a Galois biring ca

described in terms of the nature of the biring, as explained in Theorem 7.5.4. This enables it to be shown—in Corollaries 7.1 and 7.2—that a Galois ring de

moduleV for the Galois ringAG (respectivelyA〈D〉) generated by a Galois grouG (respectively Lie ringD) is, in fact, a Galois group (respectively Lie ring) descmodule forG (respectivelyD).

Let R be a Galois biring ofA with centralizerF ≡ AR . ThenGalois biring ascentbyR is simply Galois ring ascent byR—the passage from anF -spaceU to theA-moduleV ≡ A ⊗F U regarded asR-module with respect to the action ofR on V well definedby the conditionr(b ⊗ u) ≡ r(b) ⊗ u (r ∈ R,b ∈ A,u ∈ U). The resultingV is then anR-descent module withR-centralizerV R = 1⊗F U in the following sense.

Definition 7.5. An R-descent module—or Galois birings descent modulefor R on A—issimply anA-descent module forR as Galois ring ofA. And thecentralizerof R in such a
V is the centralizerV R in V of R as Galois ring ofA.


is

(1).

t

e

The simplest instance of this is the Galois birings descent moduleV = A for the Galoisbiring R of A—whoseA-module operation again is the multiplication ofA and whoseR-module operation is the endomorphism ring action ofR onA.

The Galois rings descent Theorem 7.1 leads at once to the following theorem.

Theorem 7.5 (Galois birings descent theorem). LetV be anR-descent module for a Galobiring R of A. Then

1. V R is aAR-form ofV .2. x(av) = ∑

i ix(a)xiv (x ∈ R, v ∈ V, ∆(x) = ∑i ix ⊗A xi).

Proof. ThatV R is anAR-form of V is the content of Theorem 7.1—which establishesFor (2), lettingU = V R , it suffices to prove

x(av) =∑

i

ix(a)xiv

for x ∈ R, v ∈ V , ∆(x) = ∑i ix ⊗A xi , andv = bu (b ∈ A, u ∈ U). This, in turn, is eviden

from the equations

x(av) = x(a(bu)

) = x((ab)u

) = x(ab)u =(∑

i

ix(a)xi(b)

)u =

∑i

ix(a)xi(b)u

=∑

i

ix(a)xi(bu) =∑

i

ix(a)xi(v)

for x ∈ R, v ∈ V , ∆(x) = ∑i ix ⊗A xi . �

Example 7.7. For R a Galois biring ofA and F = AR , V ≡ EndF A as A-module isan R-descent module with respect to theR-module action(rv)(b) ≡ r(v(b)) (a, b ∈ A,

r ∈ R, v ∈ V ). As in Example 7.2, its centralizer is the dual spaceV R = HomF (A,F ) =A∗ of A overF . The formula

x(ay) =∑

i

ix(a)xiy

(x, y ∈ R, ∆(x) =

∑i

ix ⊗A xi

)

follows from the second part of Theorem 7.5.

Corollary 7.1. LetG be a Galois group ofA andV a Galois rings descent module for thA-spanAG of G. ThenV is aG-descent module forA.

Proof. SinceG consists of automorphisms ofA, ∆(g) = g ⊗A g (g ∈ G). But then

g(av) = g(a)gv (g ∈ G, a ∈ A, v ∈ V )

by Theorem 7.5. SinceV is anAG-module, the nondegeneracy condition is satisfied.�


r

ining

ple

which

-

Corollary 7.2. Let D be a Galois Lie ring ofA andV a Galois rings descent module fotheA-spanA〈D〉 of the enveloping ring〈G〉 of G. ThenV is aD-descent module forA.

Proof. SinceD consists of derivations ofA, ∆(x) = x ⊗A e + e ⊗A x (x ∈ D)—wheree ≡ IA. But then

x(av) = x(a)v + axv (x ∈ D, a ∈ A, v ∈ V )

by Theorem 7.5. �

8. The simple derivation ring problem

A Lie ring D is simplewhen[D,D] �= 0 andD has no ideals other than 0 andD. WhenD is a simple Lie ring, its centroidF (centralizer ofadD in EndD) is a field—sinceD = [D,D]. So,D may be regarded as a Lie algebra overF .

In Definitions 4.1 and 4.2, a derivation ring is a Lie subringD of the Lie ringDer(A)

of derivations of some commutative ringA such thatAD ⊆ D; and when some suchA isD-simple,D is an irreducible derivation ring.

This section formulates four successively easier problems concerned with determsimple derivation rings.

Problem 8.1 (Simple derivation rings problem). Determine all simple derivation rings.

Any simple Lie ringD with centralizerF has a faithful moduleV overF—such as itsadjoint moduleD (which is faithful by the simplicity ofD). Corresponding to any suchVis itsaugmentation module algebra—theD-moduleA ≡ F ⊕ V with DF = 0 regarded ascommutative algebra overF with V being a maximal ideal ofA with V V = 0. ImbeddingD in Der(A) by way of its module action onA, D is then a simple derivation ring ofA.This proves the following theorem.

Theorem 8.1. The problem of determining all simple Lie rings is equivalent to the simderivation ring problem, Problem8.1.

Problem 8.1 becomes more tractable when a nondegeneracy condition is imposedexcludes derivation ringsD of their augmented module algebras.

Definition 8.1. A derivation ringD of A is nondegenerateif its induced action onA/M

is nonzero for every maximal idealM of A which is stable underD. A nondegeneratederivation ring is a nondegenerate derivation ring of someA.

Problem 8.2 (Simple nondegenerate derivation rings problem). Determine all simple nondegenerate derivation rings.

Theorem 8.2. A simple nondegenerate derivation ring is an irreducible derivation ring.


.

lowing

-

orems Lie

ptic

rings.r alge-n Lie

t sep-ordanion ofartic-

f Witt

llows.

Proof. Let D be a simple nondegenerate derivation ring ofA and letM be a maximalD-stable ideal ofA different fromA. SinceA is unital, such anM exists by Zorn’s lemma

Suppose thatDA ⊆ M . ThenM + Aa is a D-stable ideal ofA for a ∈ A. So, by themaximality ofM asD-stable ideal,M +Aa = A for all a ∈ A−M , that is,M is a maximalideal ofA. But the maximal idealM of A isD-stable and the action ofD onA/M is zero—contradicting the assumption thatD is a nondegenerate derivation ring ofA. This rules outDA ⊆ M .

Having ruled outDA ⊆ M , D is an irreducible derivation ring ofA ≡ A/M . After all,D �= 0, the representation ofD onA is faithful—sinceD is simple andDA �= 0; and 0,Aare the onlyD-stable ideals ofA by the choice ofM . So,D is an irreducible derivationring in the sense of Definition 4.2.�

Theorem 8.2 reduces the simple nondegenerate derivation ring problem to the folproblem.

Problem 8.3 (Simple irreducible derivation rings problem). Determine all simple irreducible derivation rings.

One direction of Problem 8.3 is solved by the following remarkably general theof [11]. This theorem establishes that all irreducible derivation rings are simple aalgebras except those of characteristic 2 which are cyclic and not “surjective.”

Theorem 8.3 (Jordan [11]). LetD be a nonzeroA-submodule and Lie subring of Der(A)

which stabilizes no ideals ofA other than0 andA. ThenD is simple as Lie algebra excepossibly when the characteristic is2 andD is cyclic asA-module. When the characteristis 2 andD = Ad , thenD is a simple Lie algebra if and only ifd(A) = A.

Theorem 8.3 reduces solving Problem 8.3 to determining all irreducible derivationOne approach to doing so, begun in [20], is based on the central simple theory fobras with operators [18]. Structure theorems [20, 7.3, 7.4] for central simple Jordaalgops and their modules are the key to the determining all simple locally nilpotenarably triangulable unital Lie algops. Classification of their closures—the simple JLie algops—then reduces to classifying those which are nil and toral. And classificattheir absolutely irreducible modules reduces to classifying those which are toral. In pular, [20, 7.3, 7.4] make tractable the problem of classifying all simple Lie algebras otype—over arbitrary fields up to purely inseparable descent.

Problem 8.3 becomes more tractable upon imposing the condition thatD be restricted.

Problem 8.4 (Simple restricted irreducible derivation rings problem). Determine all simplerestricted irreducible derivation rings.

By Theorems 4.2 and 8.3, Problem 8.4 is solved in the finitely generated case as fo

Theorem 8.4. A finitely generated restricted irreducible derivation ringD is the deriva-
tion algebra DerF A of some finite-dimensional truncated polynomial algebraA over the


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centroidF of D. It is simple if and only if the characteristic is not2 or the characteristicis 2 andA is not of the formA = F ⊕ Fa with a2 ∈ F .

Acknowledgments

This paper has benefited particularly from conversations with Bruce Allison,Faulkner, and Kevin McCrimmon following a talk on part of it given at the UniversityVirginia; from the efforts, thoughtful queries, shared insights, and suggestions of MVan den Bergh and the reviewer; and from the referenced work of Richard Block andJordan. It is a pleasure to thank them all at this time.

References

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(1981) 675–692.[14] M.E. Sweedler, Hopf Algebras, Benjamin, 1969, 336 pp.[15] D.J. Winter, Normal field extensionsK/k andK/k bialgebras, Bull. Amer. Math. Soc. 80 (3) (1974) 50

512.[16] D.J. Winter, The Structure of Fields, Grad. Texts in Math., vol. 16, Springer-Verlag, Berlin, 1974, 205[17] D.J. Winter, The Jacobson descent theorem, Pacific J. Math. 104 (2) (1983) 495–496.[18] D.J. Winter, Algops and the central simple theory, in preparation.[19] D.J. Winter, Fundamentals of Galois theory, in preparation.[20] D.J. Winter, Lie algops and simple Lie algebras, Comm. Algebra, in press.
[21] D.J. Winter, Primordial calculus and the simple primordial Lie algebras, in preparation.

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