On separable group rings

On separable group rings

Let G be a finite non-abelian group, R a ring with 1, and Ĝ the inner automorphism group of the group ring RG over R induced by the elements of G. Then three main results are shown for the separable group ring RG over R: (i) RG is not a Galois extension of (RG)Ĝ with Galois group Ĝ when the order of...

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Date:2010
Main Authors: Szeto, G., Lianyong Xue
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Published: Інститут прикладної математики і механіки НАН України 2010
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Cite this:On separable group rings / G. Szeto, Lianyong Xue // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 104–111. — Бібліогр.: 13 назв. — англ.

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spelling irk-123456789-1548822019-06-17T01:31:29Z On separable group rings Szeto, G. Lianyong Xue Let G be a finite non-abelian group, R a ring with 1, and Ĝ the inner automorphism group of the group ring RG over R induced by the elements of G. Then three main results are shown for the separable group ring RG over R: (i) RG is not a Galois extension of (RG)Ĝ with Galois group Ĝ when the order of G is invertible in R, (ii) an equivalent condition for the Galois map from the subgroups H of G to (RG)H by the conjugate action of elements in H on RG is given to be one-to-one and for a separable subalgebra of RG having a preimage, respectively, and (iii) the Galois map is not an onto map. Remove selected 2010 Article On separable group rings / G. Szeto, Lianyong Xue // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 104–111. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16S35, 16W20. http://dspace.nbuv.gov.ua/handle/123456789/154882 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description Let G be a finite non-abelian group, R a ring with 1, and Ĝ the inner automorphism group of the group ring RG over R induced by the elements of G. Then three main results are shown for the separable group ring RG over R: (i) RG is not a Galois extension of (RG)Ĝ with Galois group Ĝ when the order of G is invertible in R, (ii) an equivalent condition for the Galois map from the subgroups H of G to (RG)H by the conjugate action of elements in H on RG is given to be one-to-one and for a separable subalgebra of RG having a preimage, respectively, and (iii) the Galois map is not an onto map. Remove selected
format Article
author Szeto, G.
Lianyong Xue
spellingShingle Szeto, G.
Lianyong Xue
On separable group rings
Algebra and Discrete Mathematics
author_facet Szeto, G.
Lianyong Xue
author_sort Szeto, G.
title On separable group rings
title_short On separable group rings
title_full On separable group rings
title_fullStr On separable group rings
title_full_unstemmed On separable group rings
title_sort on separable group rings
publisher Інститут прикладної математики і механіки НАН України
publishDate 2010
url http://dspace.nbuv.gov.ua/handle/123456789/154882
citation_txt On separable group rings / G. Szeto, Lianyong Xue // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 104–111. — Бібліогр.: 13 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT szetog onseparablegrouprings
AT lianyongxue onseparablegrouprings
first_indexed 2025-07-14T06:56:47Z
last_indexed 2025-07-14T06:56:47Z
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fulltext Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 10 (2010). Number 1. pp. 104 – 111 c© Journal “Algebra and Discrete Mathematics” On separable group rings George Szeto and Lianyong Xue Communicated by V. I. Sushchansky Abstract. Let G be a finite non-abelian group, R a ring with 1, and G the inner automorphism group of the group ring RG over R induced by the elements of G. Then three main results are shown for the separable group ring RG over R: (i) RG is not a Galois extension of (RG)G with Galois group G when the order of G is invertible in R, (ii) an equivalent condition for the Galois map from the subgroups H of G to (RG)H by the conjugate action of elements in H on RG is given to be one-to-one and for a separable subalgebra of RG having a preimage, respectively, and (iii) the Galois map is not an onto map. 1. Introduction Galois extensions for rings and Hopf algebras have been intensively in- vestigated ([3], [7], [8], [10], [11]) and many examples are constructed. In [8], the following question was asked: which Azumaya algebra with an automorphism group is also a Galois algebra? In [3], it was shown that any Azumaya projective group algebra RGf over R is a central Galois algebra over R with an inner Galois group G induced by the base elements {Ug | g ∈ G} of RGf where f : G×G −→ {units of R} is a factor set ([3], Theorem 3). Recently, this fact was generalized to any separable projective group algebra RGf ([9]), and equivalent conditions were found for Galois separable skew polynomial rings and Galois crossed products with an inner This paper was written under the support of a Caterpillar Fellowship at Bradley University. The authors would like to thank Caterpillar Inc. for the support. 2000 Mathematics Subject Classification: 16S35, 16W20. Key words and phrases: Galois extensions, Galois algebras, separable extensions, group rings, group algebras. G. Szeto, L. Xue 105 Galois group ([6], [9]). The purpose of the present paper is to show that any separable group ring RG of a non-abelian group G is not a Galois extension of (RG)G with an inner Galois group G induced by the elements of G. Then we discuss the Galois map α : H −→ (RG)H by conjugation from the set of subgroups H of G to the set of separable subalgebras of RG. Also, an equivalent condition is obtained for α being one-to-one and for a separable subalgebra of RG having a preimage, respectively. Moreover, it is shown that α is not onto. 2. Basic definitions and notations Let B be a ring with 1 and A a subring of B with the same identity 1. Then B is called a separable extension of A if there exist {ai, bi in B, i = 1, 2, ..., k for some integer k} such that ∑ aibi = 1 and ∑ xai ⊗ bi = ∑ ai ⊗ bix for all x in B where ⊗ is over A. In particular, B is called an Azumaya algebra if it is a separable extension over its center ([5], Introduction or [8], Definition 2.2). Let G be a finite automorphism group of B and BG = {x ∈ B | g(x) = x for all g ∈ G}. Then B is called a Galois extension of BG with Galois group G if there exist elements {ci, di in B, i = 1, 2, ...,m for some integer m} such that ∑ cidi = 1 and ∑ cig(di) = 0 for each g 6= 1 in G. A Galois extension B of BG is called a Galois algebra if BG is contained in the center of B, and a central Galois algebra if BG is equal to the center of B. The order of a group G is denoted by |G|. Let D be a subring of B. We denote VB(D) the centralizer subring of D in B and G(D) = {g ∈ G | g(d) = d for all d ∈ D}. Let R be a ring with identity 1 and G a finite group. Then RG denotes the group ring of G over R, and RGf a projective group ring with a factor set f : G × G −→ {units in the center of R} such that f(gh, l)f(g, h) = f(g, hl)f(h, l) if RGf is a free R-module with a basis {Ug | g ∈ G} such that UgUh = f(g, h)Ugh; in particular, when R is commutative, a projective group ring RGf is called a projective group algebra. 3. Group rings of non-abelian groups In this section, let R be a ring with 1, G a finite non-abelian group of order n for some integer n invertible in R, RG the group ring of G over R, and G the inner automorphism group of RG over R induced by the elements of G. It is well known that RG is a separable extension of R. We shall show that the separable group ring RG over R is not a Galois extension of (RG)G with Galois group G. There are two cases in the proof: (i) R 106 On separable group rings is commutative and (ii) R is noncommutative. We begin with a group algebra RG over a commutative ring R. Theorem 1. Let RG be a group algebra of a finite non-abelian group over a commutative ring R and G the inner automorphism group of RG over R induced by the elements of G. Then RG is not a Galois extension of (RG)G with Galois group G. Proof. Let k be the number of conjugate classes of G and Z the center of G. Then G ∼= G/Z and |Z| < k < n for G is non-abelian where |Z| is the order of Z and n is the order of G. Let Ci be the sum of all distinct conjugate elements of the ith conjugate class of G for i = 1, 2, . . . , k, and C the center of RG. Then it is known that C = ∑k i=1RCi which is a free R-module of rank k. Now assume that RG is a Galois extension of (RG)G with Galois group G. Since (RG)G = C, RG is a central Galois algebra with an inner Galois group G. Hence RG = CGf which is a projective group algebra of G over C with a factor set f : G×G −→ {units of C} ([2], Theorem 6). Thus n = rankR(RG) = rankR(CGf ) = rankR(C) · rankC(CGf ) = k · |G| > |Z| · n/|Z| = n. This is a contradiction. Thus RG is not a Galois extension of (RG)G with Galois group G. Next, we want to extend Theorem 1 to the case of a non-commutative ring R. Let R0 be the center of R, C the center of RG, and Z the center of G. We first show some properties of G. Lemma 1. By keeping the notations in the above remarks, (1) the center of R0G is C (the center of RG) and (2) the restriction of G to R0G is isomorphic to G, that is, G|R0G ∼= G. Proof. (1) Let k be the number of conjugate classes of G and Ci the sum of all distinct conjugate elements of the ith conjugate class for i = 1, 2, . . . , k. Then C = VRG(RG) = V∑k i=1 RCi (R) = ∑k i=1R0Ci = the center of R0G. (2) Since G|R0G ∼= G/Z ∼= G, the statement holds. The following lemma which is Theorem 2.1 in [13] will play an impor- tant role. Lemma 2. Let B be a Galois extension of BG with an inner Galois group G, G = {g | g(x) = UgxU −1 g for some Ug ∈ B and for all x ∈ B}, and C the center of B. Then ∑ g∈GCUg is a projective group algebra of G over C with a factor set f : G×G −→ {units of C}. Now, we extend Theorem 1 to a separable group ring RG. G. Szeto, L. Xue 107 Theorem 2. Let RG be a group ring of a non-abelian group G of order n invertible in R, R0 the center of R, and C the center of RG. Then RG is not a Galois extension of (RG)G with an inner Galois group G induced by the elements of G. Proof. Assume that RG is a Galois extension of (RG)G with Galois group G induced by the elements of G. Then, by Lemma 2, ∑ g∈GCg = CGf which is a projective group algebra of G over C with factor set f : G × G −→ {units of C}. Since G ∼= G/Z where Z is the center of G, R0G = ∑ g∈GR0Zg ⊂ ∑ g∈GCg ⊂ ∑ g∈GR0Gg = R0G by Lemma 1. Hence R0G = ∑ g∈GCg = CGf . By Lemma 1 again, the center of R0G is C, so the center of CGf is also C. Moreover, since the order n of G is invertible in R, CGf is a separable C-algebra. Thus CGf is an Azumaya C-algebra; and so CGf is a central Galois algebra over C with an inner Galois group G ([3], Theorem 3). Therefore the group algebra R0G is a Galois algebra over C with an inner Galois group G. This contradicts to Theorem 1, so RG is not a Galois extension of (RG)G with an inner Galois group G. 4. The Galois map It is well known that the fundamental theorem holds for any indecompos- able commutative ring Galois extension S with Galois group G ([1]), that is, the Galois map α : H −→ SH for a subgroup H of G is a one-to-one correspondence between the set of subgroups of G and the set of separable subalgebras of S. Moreover, Galois extensions of a ring satisfying the fundamental theorem were studied in [12]. In this section, we shall discuss two questions of the Galois map for a non-Galois extension RG of (RG)G, α : H −→ (RG)H for a subgroup H of G where the action of H on RG is the conjugation by the elements in H: (1) when does H = G((RG)H) where G((RG)H) = {g ∈ G | g(x) = x for each x ∈ (RG)H}, that is, is α one-to-one? (2) which separable subalgebra A of RG is (RG)G(A), that is, is α onto? For a subgroup H of G, let H act on G by conjugation and Oi be the sum of all distinct conjugate elements of the ith conjugate class of G under the action of H, for i = 1, 2, . . . , h where h is the number of conjugate classes of G under the action of H. Lemma 3. By keeping the notations in the above remark, then (RG)H =∑h i=1ROi. Proof. Since H is a subgroup of G and {g | g ∈ G} is a basis for RG over R, (RG)H = ∑h i=1ROi by a direct computation. 108 On separable group rings Corollary 1. Let H and L be subgroups of G. If (RG)H = (RG)L, then (1) h = l, where h and l are the numbers of conjugate classes of G under the conjugation action of H and L respectively, and (2) {O1, O2, . . . , Oh} = {O′ 1, O ′ 2, . . . , O ′ l} where Oi is the sum of all distinct conjugate elements of the ith conjugate class of G under the action of H and O′ i is the sum of all distinct conjugate elements of the ith conjugate class of G under the action of L. Proof. (1) By Lemma 3, (RG)H = ∑h i=1ROi and (RG)L = ∑l i=1RO′ i, so ∑h i=1ROi = (RG)H = (RG)L = ∑l i=1RO′ i. Since RG is a free R- module with basis {g | g ∈ G}, (RG)H is a free R-module with basis {O1, O2, . . . , Oh} and (RG)L is a free R-module with basis {O′ 1, . . . , O ′ l}. Thus h = l. (2) Since (RG)H = (RG)L, for each i = 1, 2, . . . , h, Oi ∈ (RG)L =∑l j=1RO′ j . Hence Oi = ∑l j=1 rjO ′ j for some rj ∈ R. Noting that {g | g ∈ G} is a basis for RG over R, we have that rj is either 0 or 1. Thus Oi = ∑ j∈Ji O′ j for some subset Ji of {1, 2, . . . , l}. But {Ji | i = 1, 2, . . . , h} are disjoint subsets of {1, 2, . . . , l} where h = l, so each Ji contains only one O′ j , that is, Oi = O′ j for some j. Therefore {O1, O2, . . . , Oh} = {O′ 1, O ′ 2, . . . , O ′ l}. Now we show an equivalent condition for α being a one-to-one map: (RG)H = (RG)L implies that H = L for subgroups H and L of G. Theorem 3. Let H and L be subgroups of G. Then H = L if and only if (RG)H = (RG)L and there exists an element x ∈ G such that VH(x) = VL(x) = VHL(x) where VT (x) is the centralizer of x in T for a subset T of G. Proof. (=⇒) Since H = L, the necessity is clear. (⇐=) Since (RG)H = (RG)L, we can assume that Oi = O′ i for each i by Corollary 1. By hypothesis, there exists an element x ∈ G such that VH(x) = VL(x) = VHL(x). Since x is a term of Oi for some i and Oi = O′ i, for any a ∈ H , axa−1 = bxb−1 for some b ∈ L; and so (b−1a)x = x(b−1a), that is, b−1a ∈ VHL(x). But VL(x) = VHL(x), so b−1a ∈ L. Thus a ∈ L for any a ∈ H. This implies that H ⊂ L. Similarly, L ⊂ H. Therefore H = L. We recall that for a subset S ⊂ RG, the set {g ∈ G | g(s) = s for all s ∈ S} is denoted by G(S). Corollary 2. Let H be a subgroup of G. Then H = G((RG)H) if and only if there exists an element x ∈ G such that VG((RG)H))(x) ⊂ VH(x). G. Szeto, L. Xue 109 Proof. (=⇒) The necessity is clear. (⇐=) Since (RG)H = (RG)G((RG)H) and H ⊂ G((RG)H), the state- ment is an immediate consequence of Theorem 3. Since the order of G is invertible in R, RG is a separable group algebra over R; and so it is an Azumaya algebra over its center ([4], Example III, page 41 and Theorem 3.8, page 55). We shall show which separable subalgebra A of the Azumaya algebra RG is equal to (RG)G(A). Proposition 1. Assume the order of G is invertible in R. Then for any subgroup H of G, (RG)H is a separable R-subalgebra of RG. Proof. Let |H| = n and Tr(x) = ∑ g∈H g(x) = ∑ g∈H gxg−1. Then the map π : RG −→ (RG)H by π(x) = Tr(n−1x) is surjective as a bimodule homomorphism over (RG)H . Hence π splits. Thus (RG)H is a direct summand of RG as a bimodule over (RG)H . Since |G| is invertible in R, RG is a projective separable R-algebra. This implies that (RG)H is also a separable R-subalgebra by the proof of Theorem 3.8 in [4] on page 55. Theorem 4. Let C be the center of RG and A a separable subalgebra of the Azumaya algebra RG. Assume the order of G is invertible in R. Then A = (RG)G(A) if and only if rankCp (((RG)G(A))p) = rankCp (Ap) for each prime ideal p of C. Proof. (=⇒) The necessity is clear. (⇐=) Since RG is an Azumaya algebra over C, it is a finitely generated and projective C-module. Noting that A is a separable subalgebra of RG overC, we have that A is a direct summand of RG as an A-bimodule. Hence RG = A⊕A′ for some A-bimodule A′; and so rankCp (Ap) is defined for each prime ideal p of C ([4], page 27). Moreover, since the order of G is invertible in R, the group algebra C(G(A)) of G(A) over C is a separable subalgebra of RG over C. Thus VRG(C(G(A))) is a separable subalgebra of RG over C by the commutator theorem for Azumaya algebras ([4], Theorem 4.3, page 57). But VRG(C(G(A))) = (RG)G(A), so (RG)G(A) is a separable subalgebra of RG over C. Clearly, A ⊂ (RG)G(A), so (RG)G(A) = A⊕(A′∩ (RG)G(A)) (for RG = A ⊕ A′). By hypothesis, rankCp (((RG)G(A))p) = rankCp (Ap) for each prime ideal p of C, so rankCp ((A′ ∩ (RG)G(A))p) = 0 for each prime ideal p of C. Thus A′ ∩ (RG)G(A) = {0}. Therefore A = (RG)G(A). Next, we want to show that there are separable subalgebras of a group algebra RG of a non-abelian group G not satisfying Theorem 4, so the Galois map is not onto from the set of subgroups of G to the set of separable subalgebras of RG. 110 On separable group rings Theorem 5. Let RG be a group algebra of a non-abelian group G whose order is invertible in R. Then the Galois map α : H −→ (RG)H for a subgroup H of G is not onto from the set of subgroups of G to the set of separable subalgebras of RG. Proof. Let g 6= e (the identity of G) and R〈g〉 the subalgebra of RG generated by g. Since |G|−1 ∈ R, R〈g〉 is a separable subalgebra of RG. By hypothesis, G is non-abelian, so R〈g〉 is a proper separable subalgebra of RG. On the other hand, G(R〈g〉) = {h ∈ G |hgh−1 = g}, so G(R〈g〉) is the commutator subgroup of 〈g〉 in G. Let (RG)G(R〈g〉) = ⊕ ∑k i=1Oi where k is the number of conjugate classes of G under the action of G(R〈g〉) and Oi is the sum of the distinct conjugate elements in the ith conjugate class of G under the action. Noting that each element in 〈g〉 is a conjugate class of G under the action of G(R〈g〉) and that 〈g〉 6= G, we have that |〈g〉| < k. But rankR(R〈g〉) = |〈g〉| = the order of 〈g〉 and rankR((RG)G(R〈g〉)) = k = the number of conjugate classes of G under the action of G(R〈g〉), so R〈g〉 6= (RG)G(R〈g〉). Thus the separable subalgebra R〈g〉 does not have a preimage of α; and so the Galois map α is not onto. We conclude the present paper with an example to show that the Galois map α : H −→ (RG)H is one-to-one, but not onto. Example 1. Let S3 be the permutation group on 3 symbols {1, 2, 3, }, that is, S3 = {e, (12), (13), (23), (123), (132)}, and R the field of real numbers. Then the group algebra RS3 is not a Galois extension of (RS3) S3 with an inner Galois group S3 induced by the elements of S3, and α : H −→ (RS3) H is one-to-one from the set of subgroups of S3 to the set of separable subalgebras of the Azumaya algebra RS3 over its center C where (1) C = Re⊕R((12) + (13) + (23))⊕R((123) + (132)), (2) S3 ∼= S3 = the inner automorphism group induced by the elements of S3, (3) (RS3) 〈e〉 = RS3, (RS3) 〈(12)〉 = Re⊕R(12)⊕R((13) + (23))⊕R((123) + (132)), (RS3) 〈(13)〉 = Re⊕R(13)⊕R((12) + (23))⊕R((123) + (132)), (RS3) 〈(23)〉 = Re⊕R(23)⊕R((12) + (13))⊕R((123) + (132)), (RS3) 〈(123)〉 = Re⊕R((12) + (13) + (23))⊕R(123)⊕R(132), (RS3) S3 = C. (4) Re⊕R(12) is a separable subalgebra of RS3 which is not an image under α, so α is not onto. G. Szeto, L. Xue 111 References [1] S.U. Chase, D.K. Harrison, A. Rosenberg, Galois Theory and Galois Cohomology of Commutative Rings, Memoirs Amer. Math. Soc. No. 52, 1965. [2] F.R. DeMeyer, Some Notes on the General Galois Theory of Rings, Osaka J. Math., 2 (1965) 117-127. [3] F.R. DeMeyer, Galois Theory in Separable Algebras over Commutative Rings, Illinois J. Math., 10 (1966), 287-295. [4] F.R. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Volume 181, Springer Verlag, Berlin, Heidelberg, New York, 1971. [5] F.R. DeMeyer and G.J. Janusz, Group Rings which are Azumaya Algebras, Trans. Amer. Math. Soc., 279(1) (1983), 389-395. [6] S. Ikehata, On H-separable polynomials of prime degree, Math. J. Okayama Univ., 33 (1991), 21-26. [7] T. Kanzaki, On Galois Algebra Over A Commutative Ring, Osaka J. Math. 2 (1965), 309-317. [8] Nuss, Philippe, Galois-Azumaya Extensions and the Brauer-Galois Group of a Commutative Ring, Bull, Belg. Math. Soc., 13 (2006), 247-270. [9] G. Szeto and L. Xue, The general Ikehata theorem for H-separable crossed products, International Journal of Mathematics and Mathematical Sciences, 23(10) (2000), 657-662. [10] G. Szeto and L. Xue, The Commutator Hopf Galois Extensions, Algebra and Discrete Mathematics, 3 (2003), 89-94. [11] G. Szeto and L. Xue, The Galois Algebra with Galois Group which is the Auto- morphism Group, Journal of Algebra, 293(1) (2005), 312-318. [12] G. Szeto and L. Xue, On Galois Extensions Satisfying the Fundamental Theorem, International Mathematical Forum, 2(36) (2007), 1773-1777. [13] G. Szeto and L. Xue, On Galois Extensions with an Inner Galois Group, Recent Developments in Algebra and Related Area, ALM 8, 239-245, Higher Education Press and International Press Beijing-Boston, 2008. Contact information G. Szeto Department of Mathematics, Bradley Uni- versity, Peoria, Illinois 61625- U.S.A. E-Mail: szeto@bradley.edu L. Xue Department of Mathematics, Bradley Uni- versity, Peoria, Illinois 61625- U.S.A. E-Mail: lxue@bradley.edu Received by the editors: 04.05.2009 and in final form 04.05.2009.

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