ON CENTRAL COMMUTATOR GALOIS EXTENSIONS OF RINGS

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ON CENTRAL COMMUTATOR GALOIS EXTENSIONS OF RINGS

GEORGE SZETO and LIANYONG XUE (Received 17 November 1999)

Abstract.LetBbe a ring with 1, Ga finite automorphism group ofBof ordernfor some integern, B^Gthe set of elements inBfixed under each element inG, and∆=VB(B^G)the commutator subring ofB^GinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given.

Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofB^G if and only if B is anH-separable projective group ringB^GG_f. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.

Keywords and phrases. Azumaya algebras, Galois extensions, central Galois extensions, Azumaya Galois extensions, center Galois extensions,H-separable extensions.

2000 Mathematics Subject Classification. Primary 16S35, 16W20.

1. Introduction. Galois theory for commutative rings were studied in the sixties and seventies (see [4, Chapter 3]), and several Galois extensions of noncommutative rings were also investigated (see [2, 5, 6, 8]). Recently, central Galois extensions and the DeMeyer-Kanzaki Galois extensions were generalized to the Azumaya Galois ex- tensions and center Galois extensions, respectively (see [1, 9, 10, 11]).Bis called an Azumaya Galois extension ofB^Gwith Galois groupGifBis a Galois extension ofB^G which is an Azumaya algebra overC^G whereC is the center ofB, andBis called a center Galois extension ofB^GifCis a Galois algebra with Galois groupG|CG. The purpose of the present paper is to study a type of Galois extensions which is strictly between the types of Azumaya Galois extensions and GaloisH-separable extensions.

Let∆=VB(B^G), the commutator subring ofB^G inB. We callBa commutator Galois extension ofB^Gif∆is a Galois extension with Galois groupG|∆G, andBis a central commutator Galois extension ofB^Gif∆is a central Galois algebra with Galois group G|∆G. We shall characterize a central commutator Galois extension in terms of a GaloisH-separable extensionBofB^Gas studied by Sugano (see [8]) and theC-modules {Jg|g∈G}whereJg= {b∈B|ba=g(a)bfor alla∈B}. Moreover, it will be shown thatBis a central commutator Galois extension ofB^Gwith an inner Galois groupGif and only ifBis anH-separable projective group ringB^GGf whereB^GGf =

g∈GB^GUg

such that{Ug |g∈G}are free overB^G, bUg =Ugb for allb∈B^G and g∈G, and UgUh=Ughf (g,h)wheref:G×G→units ofC^Gis a factor set. This generalizes the structure theorem for a central Galois algebra with an inner Galois group proved by DeMeyer (see [3]).

2. Basic definitions and notation. Throughout this paper,Bwill represent a ring with 1, C the center ofB,Ga finite automorphism group ofBof ordernfor some integern,B^Gthe set of elements inBfixed under each element inG, and∆=VB(B^G), the commutator subring ofB^GinB.

LetAbe a subring of a ringBwith the same identity 1. We callBa separable extension ofAif there exist{ai,biinB, i=1,2,...,mfor some integerm}such that

aibi=1, and

bai⊗bi=

ai⊗bibfor allbinBwhere⊗is overA, and a ringBis called an H-separable extension ofAifB⊗AB is isomorphic to a direct summand of a finite direct sum of B as a B-bimodule. An Azumaya algebra is a separable extension of its center. B is called a Galois extension of B^G with Galois group G if there exist elements{ci,diinB, i=1,2,...,m}for some integermsuch that_m

i=1cig(di)=δ1,g

forg∈G. The set{ci,di}is called a G-Galois system forB. Bis called a DeMeyer- Kanzaki Galois extension ofB^GifBis an AzumayaC-algebra andCis a Galois algebra with Galois groupG|CG. IfCis a Galois algebra with Galois groupG|CG, we callB a center Galois extension ofB^G.Bis called an Azumaya Galois extension if it is a Galois extension ofB^Gthat is an AzumayaC^G-algebra, andBis called a GaloisH-separable extension if it is a Galois and an H-separable extension of B^G (see [8]). We callBa commutator Galois extension ofB^Gif∆is a Galois extension with Galois groupG|∆ G, andBis a central commutator Galois extension ofB^Gif∆is a central Galois algebra with Galois groupG|∆G. For eachg∈G, letJg= {b∈B|bx=g(x)bfor allx∈B}

andJ_g^A= {a∈A|ax=g(x)afor allx∈A}for a subringAofB.

3. Central commutator Galois extensions. In this section, we shall give several characterizations of a central commutator Galois extension in terms of Galois H- separable extensions and Azumaya Galois extensions, respectively, and prove the converse of a theorem for a GaloisH-separable extension as given in [8]. We begin with some properties of a commutator Galois extension.

Lemma3.1. IfBis a commutator Galois extension ofB^G, then∆is a Galois algebra overC^G.

Proof. Since∆is a Galois extension of∆^Gwith Galois groupG|∆G,BandB^G∆ are also Galois extensions ofB^G with Galois group G andG|B^G∆. ButB^G∆⊂B and GG|_BG∆, soB=B^G∆. Thus, the center of∆isC; and so∆^G=B^G∩∆=C^G.

Lemma3.2. IfBis a commutator Galois extension ofB^G, thenJg=Jg^∆for eachg∈G.

Proof. SinceJg= {b∈B|ba=g(a)bfor alla∈B} ⊂ {b∈B|ba=g(a)bfor all a∈B^G} =∆, Jg⊂J_g^∆.

Conversely, for anyx∈J_g^∆, xd=g(d)xfor alld∈∆. Since∆is a Galois extension of∆^G with Galois groupG|∆G, B=B^G∆by the proof of Lemma 3.1. So for any b∈B, b=_m

i=1bidifor somebi∈B^G, di∈∆and some integerm, we have thatxb= x_m

i=1bidi=_m

i=1bixdi=_m

i=1big(di)x=g_m

i=1bidi

x=g(b)x. Thus,J_g^∆⊂Jg; and soJg=J_g^∆.

Theorem3.3. The following are equivalent:

(1)Bis a central commutator Galois extension ofB^G.

(2)Bis a commutator Galois extension ofB^GandJgJg⁻¹=Cfor eachg∈G.

(3)Bis a GaloisH-separable extension ofB^G, B=B^G∆, andn⁻¹∈B.

Proof. (1)⇒(2). It is clear.

(2)⇒(1). By Lemma 3.1,∆^G=C^G, so∆is a Galois algebra with Galois groupG|∆G.

By hypothesis,JgJ_g⁻¹=Cfor eachg∈Gand by Lemma 3.2,Jg=J_g^∆for eachg∈G, so∆is a central Galois algebra (see [5, Theorem 1]).

(1)⇒(3). Since∆is a central GaloisC^G-algebra, we haveB=B^G∆, Jg=Jg^∆ for each g∈G by Lemma 3.2 andJ_g^∆J_g^∆−1=C (see [6, Lemma 2]). HenceJgJ_g−1=C for each g∈G. ButB is a Galois extension ofB^G with the same Galois system for∆, soB is a GaloisH-separable extension ofB^G (see [8, Theorem 2(iii)⇒(i)]). Moreover,n⁻¹∈B (see [6, Corollary 3]), so (3) holds.

(3)⇒(1). SinceB=B^G∆, the groupH= {g∈G|g|∆is an identity} = {1}. Thus,∆is a central Galois algebra over∆^G (see [8, Theorem 6, (3)(ii)⇒(iii)]) where∆^G=C^G by Lemma 3.1.

We remark that (1)⇒(3) in Theorem 3.3 is the converse of [8, Theorem 6]; that is, if

∆is a central Galois algebra with Galois groupG|∆G, then (i) n⁻¹∈B,

(ii) B=B^G∆,

(iii) Bis a GaloisH-separable extension ofB^G.

In the next theorem, we give a characterization of a central commutator Galois ex- tension in terms of Azumaya Galois extensions.

Theorem3.4. The following are equivalent:

(1)Bis a central commutator Galois extension ofB^GandB^Gis a separableC^G-algebra.

(2)Bis an Azumaya Galois extension with Galois groupG.

(3)Bis a central commutator Galois extension and a separable extension of∆.

Proof. (1)⇒(2). SinceBis a central commutator Galois extension,Bis a GaloisH- separable extension ofB^G by Theorem 3.3(3). Thus,VB(VB(B^G))=B^G (see [8, Propo- sition 4(1)]). This implies thatC⊂B^G; and soC=C^G. Moreover, by Theorem 3.3(3) again,B=B^G∆, so the center ofB^G isC^G, the center ofB. Thus,B^Gis an Azumaya C^G-algebra. By noting thatBis a Galois extension ofB^G, (2) holds.

(2)⇒(1). It is a consequence of [1, Lemma 1].

(1)⇒(3). SinceB is a separable extension ofB^G (for it is a Galois extension) andB^G is a separableC^G-algebra,Bis a separableC^G-algebra by the transitivity property of separable extensions. Thus,Bis a separable extension of∆becauseC^G⊂∆⊂B.

(3)⇒(1). Since∆is a Galois extension of∆^G with Galois groupG|∆G,∆is a sep- arable extension of ∆^G. By Lemma 3.1, ∆^G =C^G =C (forC is the center of∆). By hypothesis,B is a separable extension of∆. HenceB is a separable extension ofC, that is,Bis an AzumayaC-algebra. By Lemma 3.1 again,B=B^G∆such thatB^Gand∆ areC-subalgebras of the AzumayaC-algebraB. Hence, they are AzumayaC-algebras by the commutator theorem for Azumaya algebras (see [4, Theorem 4.3, page 57]).

Since∆is a Galois extension of∆^Gwith Galois groupG|∆G, Bis a Galois extension ofB^Gwhich is an AzumayaC^G-algebra. This completes the proof.

4. H-separable projective group rings. In [3], it was shown thatBis a central Galois algebra with an inner Galois groupGif and only ifBis an Azumaya projective group algebraC^GGf overC^GwhereC^GGf =

g∈GC^GUgsuch that{Ug|g∈G}are free over C^G,cUg=Ugcfor allc∈C^Gandg∈G, andUgUh=Ughf (g,h),f:G×G→units of C^Gis a factor set (see [3]). We shall generalize this fact to a central commutator Galois extension with an inner Galois group.

Theorem4.1. Bis a central commutator Galois extension ofB^Gwith an inner Galois groupGif and onlyifB=B^GGf which is anH-separable extension ofB^Gandn⁻¹∈B.

Proof. (⇒) By Theorem 3.3 (1)⇒(3),B=B^G∆which is a GaloisH-separable exten- sion ofB^Gandn⁻¹∈B, so it suffices to show thatB=B^GGf, a projective group ring with coefficient ring B^G. Since∆ is a central Galois C^G-algebra, by [3, Theorem 2],

∆=C^GGf, a projective group algebra overC^Gwheref:G×G→units ofC^Gis a factor set such thatf (g,h)=UgUhU_gh⁻¹for allg,h∈G. Noting thatbUg=Ugbfor allb∈B^G andg∈G, we claim that{Ug|g∈G}are independent overB^G. Assume

g∈GbgUg=0 for somebg∈B^G andg∈G. Since∆is a Galois extension of∆^G with Galois group G|∆G, there exists aG-Galois system{ci,di, i=1,2,...,mfor some integerm}for

∆such that_m

i=1cig(di)=δ1,gforg∈G. Hence

b1=

g∈G

δ1,gbgUg=

g∈G

m i=1

cig(di)bgUg

=

g∈G

m i=1

cibgg(di)Ug=

g∈G

m i=1

cibgUgdi

= m i=1

ci

g∈G

bgUg

di=0.

(4.1)

So

g∈GbgUg =0 for some bg ∈B^G and g∈G implies thatb1 =0. Now for any h ∈G, since

g∈GbgUg = 0, 0=

g∈GbgUgU_h⁻¹ =

g∈Gbgf (g,h⁻¹)U_gh⁻¹. Thus, bhf (h,h⁻¹)=0, and sobh=0. This proves that{Ug|g∈G}are independent overB^G. (⇐) SinceB^GGf(B^G⊗C^GC^GGf) is anH-separable extension of B^G and B^G is a direct summand ofB^GGf as a leftB^G-module,V_B^G_Gf(V_B^G_Gf(B^G))=B^G. This implies that the center of B^GGf is C^G. Moreover,G is inner induced by {Ug |g∈ G}, so Jg =C^GUg for each g∈G. But then C^GGf = ⊕

g∈GC^GUg = ⊕

g∈GJg such that JgJg⁻¹ =(C^GUg)(C^GUg⁻¹)=C^G for all g∈G. By hypothesis, n⁻¹ ∈C^G, so C^GGf

is a separable algebra overC^G. Thus,∆(=C^GGf)is a central Galois algebra (see [5, Theorem 1]) with an inner Galois group ¯Ginduced by{Ug|g∈G}. Thus,Bis a central commutator Galois extension ofB^Gwith an inner Galois groupG.

By [7, Theorem 1.2], we derive a one-to-one correspondence between some sets of separable subextensions in a central commutator Galois extensionB ofB^G. Let᏿= {Ꮽ|Ꮽis a separable subextension ofBcontainingB^Gwhich is a direct summand of Bas a bimodule}and᐀= {Ᏸ|Ᏸis a separable subalgebra of∆overC^G}.

Theorem4.2. LetB be a central commutator Galois extension ofB^G. Then, there exists a one-to-one correspondence between᏿and᐀byA→VB(A).

Proof. By Theorem 3.3(3),Bis anH-separable extension ofB^G, so the correspon- dence holds by [7, Theorem 1.2].

We conclude this paper with two examples of Galois extensionBto show that (1) Bis a central commutator Galois extension but not an Azumaya Galois exten-

sion (see Theorem 3.4),

(2) B is a GaloisH-separable extension but not a central commutator Galois ex- tension (see Theorem 3.3).

Example4.3. LetA=Q[i,j,k]be the quaternion algebra over the rational fieldQ, B=_a

1a₂

0 a3 a1,a2,a3∈A

, the ring of all 2-by-2 upper triangular matrices overA andG= {1,gi,gj,gk}wheregi(a)=iai⁻¹,gj(a)=jaj⁻¹,gk(a)=kak⁻¹for allain Aandg_a

1a₂

0 a3 =_g(a

1) g(a₂)

0 g(a3) forg∈G. Then (1)A^G=Q.

(2) B^G =_q

1q2

0 q3 q1,q2,q3 ∈Q

, the ring of all 2-by-2 upper triangar matrices overQ.

(3)∆=VB(B^G)=

a0

0a a∈A A.

(4)∆is a Galois extension of∆^G with Galois groupG|∆G and a Galois system {1,i,j,k;1/4,−i/4,−j/4,−k/4}.

(5)∆^G=Qis the center of∆.

(6) By (4) and (5),Bis a central commutator Galois extension ofB^G. (7) The center ofB^GisQ.

(8)B^G is not a separable extension of its centerQ, and soB^G is not an Azumaya algebra. In fact, suppose thatB^G is a separable extension ofQ. Then, there exists a separable idempotent

e=

1≤i≤j≤2 1≤k≤l≤2

qijkl

eij⊗ekl

, (4.2)

wheree11=

1 00 0 ,e12=

0 10 0 ,e22=

0 00 1 , andqijkl∈Qsuch that

1≤i≤j≤2 1≤k≤l≤2

qijkleijekl=I2, (4.3)

the identity 2-by-2 matrix, andbe=ebfor allb∈B^G. Bye11e=ee11, we have

1≤j≤2 1≤k≤l≤2

q1jkl

e1j⊗ekl

=

1≤i≤j≤2

qij11

eij⊗e11

. (4.4)

Henceq2211=0 andq1jk2=0 for allj,k, that is,q1112=q1122=q1212=q1222=0. By e12e=ee12, we have

1≤k≤l≤2

q22kl

e12⊗ekl

=

1≤i≤j≤2

qij11

eij⊗e12

. (4.5)

Henceq22kl=0 if(k,l)=(1,2)andqij11=0 if(i,j)=(1,2), that is,q2211=q2222=0 andq1111=q2211=0. Therefore,e=q1211(e12⊗e11)+q2212(e22⊗e12). Thus,

I2=

1≤i≤j≤2 1≤k≤l≤2

qijkleijekl=q1211e12e11+q2212e22e12=0. (4.6)

This contradiction shows thatB^Gis not a separable extension ofQ.

Example4.4. LetB=Q[i,j,k]be the quaternion algebra over the rational fieldQ andG= {1,gi}wheregi(x)=ixi⁻¹for allxinB. Then

(1)Bis a Galois extension ofB^Gwith Galois groupGand a Galois system{1,i,j,k;1/4,

−i/4,−j/4,−k/4}.

(2) SinceGis inner,Bis anH-separable extension ofB^G. (3) By (1) and (2),Bis a GaloisH-separable extension ofB^G.

(4)∆=VB(B^G)=Q[i]is not a Galois extension of∆^G with Galois groupG|∆G, and soBis not a central commutator Galois extension ofB^G.

Acknowledgement. This paper was written under the support of a Caterpil- lar Fellowship at Bradley University. We would like to thank Caterpillar Inc. for the support.

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George Szeto: Mathematics Department, Bradley University, Peoria, Illinois61625, USA

E-mail address:[email protected]

Lianyong Xue: Mathematics Department, Bradley University, Peoria, Illinois61625, USA

E-mail address:[email protected]