ON CHARACTERIZATIONS OF A CENTER GALOIS EXTENSION

Vol. 23, No. 11 (2000) 753–758 S0161171200003562

© Hindawi Publishing Corp.

ON CHARACTERIZATIONS OF A CENTER GALOIS EXTENSION

GEORGE SZETO and LIANYONG XUE (Received 16 June 1999)

Abstract.LetBbe a ring with 1,Cthe center ofB,Ga finite automorphism group ofB, andB^Gthe set of elements inBfixed under each element inG. Then, it is shown thatBis a center Galois extension ofB^G(that is,Cis a Galois algebra overC^Gwith Galois group G|CG) if and only if the ideal ofBgenerated by{c−g(c)|c∈C}isBfor eachg≠1 in G. This generalizes the well known characterization of a commutative Galois extensionC thatCis a Galois extension ofC^Gwith Galois groupGif and only if the ideal generated by {c−g(c)|c∈C}isCfor eachg≠1 inG. Some more characterizations of a center Galois extensionBare also given.

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

2000 Mathematics Subject Classification. Primary 16S30, 16W20.

1. Introduction. LetCbe a commutative ring with 1,Ga finite automorphism group ofCandC^Gthe set of elements inCfixed under each element inG. It is well known that a commutative Galois extensionC is characterized in terms of the ideals generated by{c−g(c)|c∈C}forg≠1 inG, that isCis a Galois extension with Galois groupG if and only if the ideal generated by{c−g(c)|c∈C}isC for eachg≠1 inG(see [3, Proposition 1.2, page 80]). A natural generalization of a commutative Galois extension is the notion of a center Galois extension, that is, a noncommutative ringBwith a finite automorphism groupGand centerCis called a center Galois extension ofB^G with Galois groupGifC is a Galois extension ofC^G with Galois group G|C G. Ikehata (see [4, 5]) characterized a center Galois extension with a cyclic Galois group G of prime order in terms of a skew polynomial ring. Then, the present authors generalized the Ikehata characterization to center Galois extensions with Galois groupG of any cyclic order [7] and to center Galois extensions with any finite Galois groupG [8].

The purpose of the present paper is to generalize the above characterization of a commutative Galois extension to a center Galois extension. We shall show thatBis a center Galois extension ofB^Gif and only if the ideal ofBgenerated by{c−g(c)|c∈C}

isBfor each g≠1 in G. A center Galois extensionBis also equivalent to each of the following statements:

(i) Bis a Galois central extension ofB^G, that is,B=B^GCwhich isG-Galois extension ofB^G.

(ii) Bis a Galois extension ofB^Gwith a Galois system{bi∈B, ci∈C, i=1,2,...,m}

for some integerm.

(iii) the ideal of the subringB^GC generated by{c−g(c)|c∈C}isB^GC for each g≠1 inG.

2. Definitions and notations. Throughout this paper,Bwill represent a ring with 1,G= {g1=1, g2,...,gn}an automorphism group ofBof ordernfor some integer n,Cthe center ofB, B^Gthe set of elements inBfixed under each element inG, and B∗Ga skew group ring in which the multiplication is given bygb=g(b)gforb∈ Bandg∈G.

Bis called aG-Galois extension ofB^Gif there exist elements{ai,bi∈B, i=1,2,...,m}

for some integermsuch that_m

i=1aig(bi)=δ1,g. Such a set{ai,bi}is called aG-Galois system forB.Bis called a center Galois extension ofB^G ifCis a Galois algebra over C^Gwith Galois groupG|CG.Bis called a central extension ofB^GifB=B^GC, andB is called a Galois central extension ofB^GifB=B^GCis a Galois extension ofB^Gwith Galois groupG.

Let A be a subring of a ring B with the same identity 1. We denote VB(A) the commutator subring of Ain B. We callB a separable extension of Aif there exist {ai,bi∈B, i=1,2,...,mfor some integerm}such that

aibi=1,and

bai⊗bi= ai⊗bibfor allb∈Bwhere⊗is overA.Bis calledH-separable extension ofAif B⊗ABis isomorphic to a direct summand of a finite direct sum ofBas aB-bimodule.

Bis called centrally projective overAifBis a direct summand of a finite direct sum ofAas aA-bimodule.

3. The characterizations. In this section, we denoteJ_j^(C)= {c−gj(c)|c∈C}. We shall show thatBis a center Galois extension ofB^Gif and only ifB=BJ_j^(C), the ideal ofBgenerated byJ_j^(C),for eachgj≠1 inG. Some more characterizations of a center Galois extensionBare also given. We begin with a lemma.

Lemma3.1. IfB=BJ_j^(C)for eachgj≠1inG(that is,j≠1), then

(1)Bis a Galois extension ofB^Gwith Galois groupGand a Galois system{bi∈B;ci∈ C, i=1,2,...,m}for some integerm.

(2)Bis a centrally projective overB^G. (3)B∗GisH-separable overB.

(4)VB∗G(B)=C.

Proof. (1) SinceB =BJ_j^(C) for each j ≠1, there exist

b^(j)_i ∈B, c_i^(j) ∈C, i= 1,2,...,mj

for some integermj, j=2,3,...,nsuch thatmj i=1b_i^(j)

c_i^(j)−gj c_i^(j)

=1.

Therefore, mj

i=1b^(j)_i c_i^(j) =1+mj i=1b^(j)_i gj

c^(j)_i

. Letb^(j)_mj+1= −mj i=1b^(j)_i gj

c_i^(j) and c_m^(j)_j+1=1. Thenmj+1

i=1 b^(j)_i c_i^(j)=1 andmj+1 i=1 b^(j)_i gj

c_i^(j)

=0. Letbi2,i3,...,in=b⁽²⁾_i2 b_i⁽³⁾₃

···b⁽ⁿ⁾_in andci2,i3,...,in=c⁽²⁾_i2 c_i⁽³⁾₃ ...c_i⁽ⁿ⁾_n forij=1,2,...,mj+1 andj=2,3,...,n. Then

m2+1 i2=1

m3+1 i3=1

···

mn+1 in=1

bi2,i3,...,inci2,i3,...,in=

m2+1 i2=1

m3+1 i3=1

···

mn+1 in=1

b_i⁽²⁾₂ b⁽³⁾_i3 ···b_i⁽ⁿ⁾_n c⁽²⁾_i2 c_i⁽³⁾₃ ···c⁽ⁿ⁾_in

=

m₂+1 i2=1

m₃+1 i3=1

···

mn+1 in=1

b⁽²⁾_i2 c_i⁽²⁾₂ b⁽³⁾_i3 c⁽³⁾_i3 ···b⁽ⁿ⁾_in c_i⁽ⁿ⁾_n

=

m2+1 i2=1

b⁽²⁾_i2 c⁽²⁾_i2

m3+1 i3=1

b⁽³⁾_i3 c⁽³⁾_i3 ···

mn+1 in=1

b_i⁽ⁿ⁾_n c_i⁽ⁿ⁾_n =1 (3.1)

and for eachj≠1

m2+1 i2=1

m3+1 i3=1

···

mn+1 in=1

bi2,i₃,...,ingj(ci₂,i₃,...,in)

=

m2+1 i2=1

m₃+1 i3=1

···

mn+1 in=1

b_i⁽²⁾₂ b⁽³⁾_i3 ···b_i⁽ⁿ⁾_n gj

c⁽²⁾_i2 c_i⁽³⁾₃ ···c_i⁽ⁿ⁾_n

=

m2+1 i2=1

m3+1 i3=1

···

mn+1 in=1

b_i⁽²⁾₂ b⁽³⁾_i3 ···b_i⁽ⁿ⁾_n gj

c⁽²⁾_i2 gj

c_i⁽³⁾₃

···gj

c_i⁽ⁿ⁾_n

=

m2+1 i₂=1

m3+1 i₃=1

···

mn+1 in=1

b_i⁽²⁾₂ gj

c_i⁽²⁾₂ b_i⁽³⁾₃ gj

c_i⁽³⁾₃

···b⁽ⁿ⁾_in gj

c_i⁽ⁿ⁾_n

=

m2+1 i2=1

b⁽²⁾_i2 gj

c⁽²⁾_i2 ^m3+1

i3=1

b⁽³⁾_i3 gj

c⁽³⁾_i3

···

mn+1 in=1

b⁽ⁿ⁾_in gj

c_i⁽ⁿ⁾_n

=0.

(3.2)

Thus,{bi₂,i₃,...,in∈B; ci₂,i₃,...,in∈C, ij=1,2,...,mj+1 andj=2,3,...,n}is a Galois system forB. This complete the proof of (1).

(2) By (1),B is a Galois extension ofB^G with a Galois system {bi∈B, ci∈C, i= 1,2,...,m}for some integerm. Letfi:B→B^Ggiven byfi(b)=_n

j=1gj(cib)for allb∈ B, i=1,2,...,m. Then it is easy to check thatfiis a homomorphism asB^G-bimodule and b=_m

i=1bicib =_n

j=1_m

i=1bigj(ci)gj(b)=_m

i=1bi_n

j=1gj(cib)=_m

i=1bifi(b) for allb∈B. Hence{bi;fi, i=1,2,...,m}is a dual bases forBasB^G-bimodule, and so Bis finitely generated and projective asB^G-bimodule. Therefore,Bis a direct summand of a finite direct sum ofB^Gas aB^G-bimodule. ThusBis centrally projective overB^G.

(3) By (1),Bis a Galois extension ofB^Gwith Galois groupG. HenceB∗GHomB^G(B, B)[2, Theorem 1]. By (2),Bis centrally projective overB^G. Thus,B∗G(HomB^G(B,B)) isH-separable overB[6, Proposition 11].

(4) We first claim thatVB∗G(C)=B. Clearly,B⊂VB∗G(C). Let_n

j=1bjgjin VB∗G(C) for somebj∈B. Thenc(_n

j=1bjgj)=(_n

j=1bjgj)cfor eachc∈ C, socbj=bjgj(c), that is,bj(c−gj(c))=0 for eachgj∈Gandc∈C. SinceB=BJ_j^(C)for eachgj≠1, there existb^(j)_i ∈B andc_i^(j)∈C, i=1,2,...,msuch that_m

i=1b^(j)_i

c_i^(j)−gj c_i^(j)

= 1. Hence bj=_m

i=1b^(j)_i

c_i^(j)−gj c_i^(j)

bj=_m

i=1b_i^(j)bj

c_i^(j)−gj c_i^(j)

=0 for each gj≠1. This implies that_n

j=1bjgj=b1∈B. HenceVB∗G(C)⊆B, and soVB∗G(C)=B.

Therefore,VB∗G(B)⊂VB∗G(C)=B. ThusVB∗G(B)=VB(B)=C.

We now show some characterizations of a center Galois extensionB.

Theorem3.2. The following statements are equivalent.

(1)Bis a center Galois extension ofB^G. (2)B=BJ_j^(C)for eachgj≠1inG.

(3)Bis a Galois extension ofB^Gwith a Galois system{bi∈B, ci∈C, i=1,2,...,m}

for some integerm.

(4)Bis a Galois central extension ofB^G. (5)B^GC=B^GCJ_j^(C)for eachgj≠1inG.

Proof. (1)⇒(2). By hypothesis,C is a Galois extension ofC^G with Galois group G|C G. HenceC=CJ_j^(C)for eachgj≠1 in G[3, Proposition 1.2, page 80]. Thus, B=BJ_j^(C)for eachgj≠1 inG.

(2)⇒(1). SinceB=BJ_j^Cfor eachgj≠1 inG, B∗GisH-separable overBby Lemma 3.1(3) and VB∗G(B)=C by Lemma 3.1(4). Thus C is a Galois extension ofC^G with Galois groupG|CGby [1, Proposition 4].

(1)⇒(3). This is Lemma 3.1(1).

(3)⇒(1). SinceBis Galois extension ofB^Gwith a Galois system{bi∈B, ci∈C, i= 1,2,...,m}for some integerm, we have_m

i=1bigj(ci)=δ1,g. Hence_m

i=1bi(ci−gj(ci))

=1 for eachgj≠1in G. So for everyb∈B,b=_m

i=1bbi(ci−gj(ci))∈BJ^(C)_j . There- fore,B=BJ_i^(C)for each gi≠1in G. Thus,B is a center Galois extension ofB^G by (2)⇒(1).

(1)⇒(4). SinceCis a Galois algebra with Galois groupG|CG,BandB^GCare Galois extensions ofB^Gwith Galois groupG|_B^G_CG. Noting thatB^GC⊂B, we haveB=B^GC, that is,Bis a central extension ofB^G. ButBis a Galois extension ofB^G, soBis a Galois central extension ofB^G.

(4)⇒(1). By hypothesis,B=B^GC is a Galois extension ofB^G. Hence there exists a Galois system{ai;bi∈B, i=1,2,...,m}for some integermsuch that_m

i=1aigj(bi)= δ1,j. ButB=B^GC, soai=n_ai

k=1b^(a_kⁱ⁾c^(a_kⁱ⁾andbi=ⁿ_bi

l=1b_l^(bi)c_l^(bi)for somea^(a_kⁱ⁾, b^(b_l ⁱ⁾ inB^Gandc_k^(ai), c^(b_l ⁱ⁾in C, k=1,2...,na_i, l=1,2,...,nb_i, i=1,2,...,m. Therefore,

δ1,j= m i=1

aigj(bi)= m i=1

n_ai

k=1

b^(a_kⁱ⁾c_k^(ai)gj n_bi

l=1

b^(b_l ⁱ⁾c_l^(bi)

= m i=1

n_ai

k=1

b_k^(ai)c_k^(ai)

n_bi

l=1

b^b_lⁱgj

c^(b_l ⁱ⁾

= m i=1

n_ai

k=1 n_bi

l=1

b^(a_kⁱ⁾c_k^(ai)b^(b_l ⁱ⁾ gj

c_l^(bi) .

(3.3)

This shows that

b_k,l^(ai,bi)=b^(a_kⁱ⁾c_k^(ai)b_l^(bi)∈B; c_k,l^(ai,bi)=c_l^(bi)∈C, k=1,2,...,nai, l= 1,2,...,nb_i, i=1,2,...,m

is a Galois system forB. Thus,Bis a center Galois extension ofB^Gby (3)⇒(1).

(1)⇒(5). SinceBis a center Galois extension ofB^G,B=BJ^(C)_j for eachgj≠1 inG by (1)⇒(2) andB=B^GCby (1)⇒(4). Thus, B^GC=B^GCJ^(C)_j for eachgj≠1 inG.

(5)⇒(1). SinceB^GC=B^GCJ_j^(C)for eachgj≠1 inG,B=BJ_j^(C)for eachgj≠1 inG.

Thus,Bis a center Galois extension ofB^Gby (2)⇒(1).

The characterization of a commutative Galois extensionC in terms of the ideals generated by {c−g(c)| c ∈C} for g ≠1 in G is an immediate consequence of Theorem 3.2.

Corollary3.3. A commutative ringC is a Galois extension ofC^G if and only if C=CJ_j^(C), the ideal generated by{c−gj(c)|c∈C}isCfor eachgj≠1inG.

Proof. LetB=Cin Theorem 3.2. Then, the corollary is an immediate consequence of Theorem 3.2(2).

By Theorem 3.2, we derive several characterizations of a Galois centeral extensionB.

Corollary3.4. IfBis a central extension ofB^G(that is,B=B^GC), then the following statements are equivalent.

(1)Bis a Galois extension ofB^G. (2)Bis a center Galois extension ofB^G. (3)B∗GisH-separable overB.

(4)B=CJ_j^(B)for eachgj≠1inG.

(5)B=BJ_j^(B)for eachgj≠1inG.

Proof. (1)⇐⇒(2). This is given by (1)⇐⇒(4) in Theorem 3.2.

(2)⇒(3). This is Lemma 3.1(3).

(3)⇒(1). SinceB∗GisH-separable overB,Bis a Galois extension ofB^G[1, Propo- sition 2].

SinceB=B^GCby hypothesis, it is easy to see thatJ_j^(B)=B^GJ_j^(C)for eachgjinG. Thus, B=CJ_j^(B), B=BJ^(B)_j , andB=BJ_j^(C)are equivalent. This implies that (2)⇐⇒(4)⇐⇒(5) by Theorem 3.2(2).

We call a ringBthe DeMeyer-Kanzaki Galois extension ofB^GifBis an AzumayaC- algebra andBis a center Galois extension ofB^G(for more about the DeMeyer-Kanzaki Galois extensions, see [2]). Clearly, the class of center Galois extensions is broader than the class of the DeMeyer-Kanzaki Galois extensions. We conclude the present paper with two examples. (1) The DeMeyer-Kanzaki Galois extension ofB^G and (2) a center Galois extension ofB^G, but not the DeMeyer-Kanzaki Galois extension ofB^G.

Example3.5. LetCbe the field of complex numbers, that is, C=R+R√

−1 where Ris the field of real numbers,B=C[i,j,k]the quaternion algebra overC, andG= {1,g|g(c1+cii+cjj+ckk)=g(c1)+g(ci)i+g(cj)j+g(ck)kfor eachb=c1+cii+

cjj+ckk∈C[i,j,k]andg(u+v√

−1)=u−v√

−1 for eachc=u+v√

−1∈C}. Then (1) The center ofBisC.

(2)Bis an AzumayaC-algebra.

(3)Cis a Galois extension ofC^G with Galois groupG|CGand a Galois system {a1=1/√

2, a2=(1/√ 2)√

−1; b1=1/√

2, b2= −(1/√ 2)√

−1}.

(4)Bis the DeMeyer-Kanzaki Galois extension ofB^Gby (2) and (3).

(5)B^G=R[i,j,k].

(6)B=B^GC, soBis a centeral extension ofB^G. (7)J^(C)g =R√

−1.

(8)B=BJg^(C)since 1= −√

−1√

−1∈BJg^(C). (9)J^(B)g =R√

−1+R√

−1i+R√

−1j+R√

−1k.

(10)B=CJg^(B).

Example3.6. By replacing in Example 3.5 the field of complex numbers Cwith the ringC=Z⊕ZwhereZis the ring of integers,g(a,b)=(b,a)for all(a,b)∈C, andG= {1,g|g(c1+cii+cjj+ckk)=g(c1)+g(ci)i+g(cj)j+g(ck)kfor eachb= c1+cii+cjj+ckk∈B=C[i,j,k]}. Then

(1) The center ofBisC.

(2)C is a Galois extension ofC^G with Galois groupG|C G and a Galois system {a1=(1,0), a2=(0,1); b1=(1,0), b2=(0,1)}.

(3)Bis not an AzumayaC-algebra (for 1/2∉C), and soBis not the DeMeyer-Kanzaki Galois extension ofB^G.

(4)C^G= {(a,a)|a∈Z} Z.

(5)B^G=C^G[i,j,k].

(6)B=B^GC, soBis a central extension ofB^G. (7)J^(C)g = {(a,−a)|a∈Z} =Z(1,−1).

(8)B=BJg^(C)since 1=(1,1)=(1,−1)(1,−1)∈BJ_G^(C). (9)J^(B)g =Z(1,−1)+Z(1,−1)i+Z(1,−1)j+Z(1,−1)k.

(10)B=CJg^(B).

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