����������� ������ 52 (2000), 63–69

UDK 512.552

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ON SPLITTING RINGS FOR AZUMAYA SKEW GROUP RINGS

George Szeto and Lianyong Xue

Abstract. Let B be a ring with 1, G an automorphism group of B of order n for some integer n, B ∗ G the skew group ring over B with a free basis {g | g ∈ G}, B^{G} the set of elements in B fixed under G, and \overline{G} the inner automorphism group of B ∗ G induced by G. It is shown that when the center C of B is a G-Galois algebra over C^{G} with Galois group G|_{C} ≅ G or B is a G-Galois extension of B^{G} and n^{−1} ∈ B, then, B ∗ G is an Azumaya algebra if and only if so is (B ∗ G)\overline{G}, and some splitting rings of B ∗ G, (B ∗ G)\overline{G} and B are shown to be the same.

1. Introduction

Let B be a ring with 1, C the center of B, G an automorphism group of B of order n for some integer n, B ∗ G a skew group ring over B with a free basis \{ g | g ∈ G \}, B^{G} the set of elements in B fixed under G, \overline{G} the inner automorphism group of B ∗ G induced by G, that is, ḡ(f) = gfg^{−1} for each f ∈ B ∗ G and g ∈ G. We note that \overline{G} restricted to B is G.

In [1] and [2], the Azumaya skew group ring B∗G over C^{G} was characterized in terms of Azumaya Galois extension B of B^{G} and the H-separable extension B ∗ G of B respectively. Also in [3], the commutator subring of B in B ∗ G was studied. In the present paper, under a Galois condition on B, the Azumaya skew group ring B ∗ G is characterized in terms of the Azumaya fixed subring (B ∗ G)\overline{G} under \overline{G} and the Azumaya coefficient ring B, that is, when C is a G-Galois algebra over C^{G} with Galois group G|_{C} ≅ G or B is a G-Galois extension of B^{G} and n^{−1} ∈ B, then, B ∗ G is an Azumaya algebra if and only if so is (B ∗ G)\overline{G}.

Let A be an Azumaya algebra. It is well known that any separable maximal commutative subalgebra of A is a splitting ring for A ([4], Theorem 5.5, p. 64). In this paper, we call F a splitting ring for A if F is a separable maximal commutative subalgebra of A. We then show that when C is a G-Galois algebra over C^{G} with Galois group G|_{C} ≅ G, F is a splitting ring for the Azumaya algebra B ∗ G containing C if and only if F is a splitting ring for the Azumaya algebra B. Moreover, when B is a G-Galois extension of B^{G} and n^{−1} ∈ B, F is a splitting ring for the Azumaya algebra B ∗G containing the center of (B ∗G)\overline{G}, then, F is a splitting ring for (B ∗ G)\overline{G} if and only if G is Abelian. At the end, two examples are constructed to demonstrate the results.

AMS Subject Classification: 16 S 30, 16 W 20

Keywords and phrases: Skew group rings, Azumaya algebras, Galois extensions, splitting rings

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

2. Basic definitions and notations

Throughout this paper, B will represent a ring with 1, G an automorphism group of B, C the center of B, B ∗ G a skew ring in which the multiplication is given by gb = g(b)g for b ∈ B and g ∈ G, B^{G} the set of elements in B fixed under G, Z the center of B ∗ G, \overline{G} the inner automorphism group of B ∗ G induced by G, that is, ḡ(f) = gfg^{−1} for each f ∈ B ∗ G and g ∈ G. We note that \overline{G} restricted to B is G.

Let A be a subring of a ring B with the same identity 1. We denote V_{B}(A) the commutator subring of A in B. We call B a separable extension of A if there exist { a_{i}, b_{i} in B, i = 1, 2,…, m for some integer m } such that ∑ a_{i}b_{i} = 1, and ∑ ba_{i} ⊗ b_{i} = ∑ a_{i} ⊗ b_{i}b for all b in B where ⊗ is over A, and a ring B is called a H-separable extension of A if B ⊗_{A} B 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 G-Galois extension of B^{G} if there exist elements { c_{i}, d_{i} in B, i = 1, 2,…, m } for some integer m such that ∑_{i=1}^{m} c_{i}g(d_{i}) = δ_{1,g}. The set {c_{i}, d_{i}} is called a G-Galois system for B. B is called a DeMeyer-Kanzaki G-Galois extension if B is an Azumaya C-algebra and C is G-Galois algebra with G|_{C} ≅ G. If A is an Azumaya C-algebra and S is a commutative C-algebra such that A ⊗_{C} S ≅ Hom_{S}(E, E) for some S-progenerator E, then S is called a splitting ring for the Azumaya algebra A. It is well known (Theorem 5.5 in [4] on p. 64) that any separable maximal commutative subalgebra of A is a splitting ring for A. In the present paper, S is called a splitting ring for A if S is a separable maximal commutative subalgebra of A.

3. Characterizations of Azumaya skew group rings

In this section we shall characterize an Azumaya skew group ring B∗G in terms of (B ∗ G)\overline{G} and B under a Galois condition that C is a G-Galois algebra over C^{G} with Galois group G|_{C} ≅ G or B is a G-Galois extension of B^{G} and n^{−1} ∈ B. We begin with a Lemma.

Lemma 3.1. If C is a G-Galois algebra over C^{G} with Galois group G|_{C} ≅ G, then

(a) B ∗ G is H-separable over B.

(b) B ∗ G is H-separable over (B ∗ G)\overline{G}.

(c) The center of B ∗ G, Z = C^{G}.

(d) V_{B∗G}(C) = B.

Proof. (a) Since C is a G-Galois algebra over C^{G} with Galois group G|_{C} ≅ G and C ⊆ V_{B∗G}(B), V_{B∗G}(B) is \overline{G}-Galois extension of (V_{B∗G}(B))\overline{G} with the same Galois system as C. Hence, B∗G is H-separable extension of B by ([3], Theorem 1).

(b) Since C is a G-Galois extension of C^{G} with Galois group G|_{C} ≅ G, B ∗G is a \overline{G}-Galois extension of (B ∗ G)\overline{G} with the same Galois system as C. But \overline{G} acts on B ∗ G is inner, so B ∗ G is H-separable extension of (B ∗ G)\overline{G} by ([7], Corollary 3).

(c) By (a), B ∗ G is H-separable over B. Moreover, B is a direct summand of B ∗ G as a left B-module, so B satisfies the double centralizer property in B ∗ G ([8], Proposition 1.2), that is, B = V_{B∗G}(V_{B∗G}(B)). This implies that the center of B ∗ G is contained in B. Thus, Z = C^{G}.

(d) Clearly, B ⊆ V_{B∗G}(C). Conversely, for each ∑_{g∈G} b_{g}g in V_{B∗G}(C), we have c(∑_{g∈G} b_{g}g) = (∑_{g∈G} b_{g}g)c for each c in C, so cb_{g} = b_{g}g(c), that is, b_{g}(c−g(c)) = 0 for each g ∈ G and c ∈ C. But C is a commutative G-Galois extension of C^{G}, so the ideal of C generated by { c − g(c) | c ∈ C } is C ([4], Proposition 1.2-(5)). Thus b_{g} = 0 for each g ≠ 1. But then ∑_{g∈G} b_{g}g = b_{1} ∈ B. Hence V_{B∗G}(C) ⊆ B, and so V_{B∗G}(C) = B.

Theorem 3.2. Assume C is a G-Galois algebra over C^{G} with Galois group G|_{C} ≅ G. The the following statements are equivalent:

(1) B ∗ G is Azumaya.

(2) (B ∗ G)\overline{G} is Azumaya.

(3) B is Azumaya.

Proof. (1) ⟺ (2). Since C is a G-Galois algebra over C^{G} with Galois group G|_{C} ≅ G, there exists an element c ∈ C such that Tr_{G}(c) = 1, where Tr_{G}( ) is the trace of G ([4], Corollary 1.3-(1)). By Lemma 3.1-(b), B ∗ G is H-separable over (B ∗ G)\overline{G} and B ∗ G is a finitely generated and projective left module over (B ∗ G)\overline{G} ([5], Theorem 1), so (2) ⟹ (1) by ([6], Theorem 1). Conversely, since the restriction of \overline{G} to C is G, (B ∗G)\overline{G} is a direct summand of B ∗G as a (B ∗G)\overline{G}- bimodule by using the fact that Tr_{G}(c) = 1. Thus the separability of B ∗ G over Z implies the separability of (B ∗ G)\overline{G} over Z by the argument as given on p. 120 in [5]. Since Z is contained in the center of (B ∗ G)\overline{G}, (B ∗ G)\overline{G} is Azumaya. This proves (1) ⟹ (2).

(1) ⟹ (3). Assume B ∗ G is Azumaya. Since C is a G-Galois algebra over C^{G}, Z = C^{G} by Lemma 3.1-(c). Hence B ∗ G is an Azumaya C^{G}-algebra. By Lemma 3.1-(d), V_{B∗G}(C) = B. Therefore, B is a separable C^{G}-algebra (for C is a separable C^{G}-algebra) by the commutator theorem for Azumaya algebras ([4], Theorem 4.3, p. 57). Thus B is an Azumaya algebra.

(3) ⟹ (1). Since C is a G|_{C}-Galois algebra over C^{G}, B ∗ G is a H-separable extension of B by Lemma 3.1-(a). By hypothesis, B is an Azumaya C-algebra, so B ∗ G is a separable extension over C by the transitivity of separable extensions. Noting that C is a separable C^{G}-algebra (for it is G-Galois), we conclude that B ∗G is a separable extension of C^{G}. Moreover, by Lemma 1-(c), Z = C^{G}, so B ∗ G is an Azumaya C^{G}-algebra.

Theorem 3.3. Let B be a G-Galois extension of B^{G} and n^{−1} ∈ B. Then, B ∗ G is an Azumaya algebra if and only if so is (B ∗ G)\overline{G}. In this case, the center of (B ∗ G)\overline{G} is the center of ZG where Z is the center of B ∗ G.

Proof. Since n^{−1} ∈ B, Tr_{G}(n^{−1}) = 1. By hypothesis B is a G-Galois extension of B^{G}, so B ∗ G is a \overline{G}-Galois extension of (B ∗ G)\overline{G} with an inner Galois group \overline{G} with the same Galois system as B. Thus the argument in the proof of (1) ⟺ (2) in Theorem 3.2 implies that B∗G is an Azumaya algebra if and only if so is (B∗G)\overline{G}.

Next, we calculate the center of (B ∗ G)\overline{G}. Let Z be the center of B ∗ G. Then the center of (B ∗ G)\overline{G} = V_{(B∗G)\overline{G}}((B ∗ G)\overline{G}) = (B ∗ G)\overline{G} ∩ V_{B∗G}((B ∗ G)\overline{G}) =

(B ∗ G)\overline{G} ∩ V_{B∗G}(V_{B∗G}(ZG)). Since n^{−1} ∈ B, ZG is a separable Z-algebra. Hence V_{B∗G}(V_{B∗G}(ZG)) = ZG because B ∗G is an Azumaya Z-algebra ([4], Theorem 4.3, p. 57). Thus, the center of (B ∗ G)\overline{G} = (B ∗ G)\overline{G} ∩ (ZG) = V_{B∗G}(ZG) ∩ (ZG) = V_{ZG}(ZG) = the center of ZG.

4. Splitting rings

In this section, we shall show that some splitting rings for B ∗ G, (B ∗ G)\overline{G} and B are the same. Recall that a splitting ring is a separable maximal commutative subalgebra. We first give a result on the splitting rings for any Azumaya algebra.

Theorem 4.1. Let A be an Azumaya C-algebra and D a separable commutative subalgebra of A. Then (i) V_{A}(D) is an Azumaya D-algebra, and (ii) F is a splitting ring for A containing D if and only if F is a splitting ring for V_{A}(D) over D.

Proof. (i) Since A is an Azumaya C-algebra and D a separable subalgebra of A, V_{A}(V_{A}(D)) = D and V_{A}(D) is separable subalgebra of A by the commutator theorem for Azumaya algebras ([4], Theorem 4.3, p. 57). Since D is a commutative subalgebra of A, C ⊂ D ⊂ the center of V_{A}(D). Hence V_{A}(D) is separable over D. Moreover, the center of V_{A}(D) = V_{V_{A}(D)}(V_{A}(D)) ⊂ V_{A}(V_{A}(D)) = D; and so the center of V_{A}(D) = D, that is, V_{A}(D) is an Azumaya D-algebra.

(ii) ( ⟹ ) Let F be a splitting ring for A containing D. Then D ⊂ F and F = V_{A}(F), and so F = V_{A}(F) ⊂ V_{A}(D). Hence V_{V_{A}(D)}(F) = V_{A}(D) ∩ V_{A}(F) = V_{A}(F) = F. Thus F is a maximal commutative subalgebra of V_{A}(D). Moreover, since F is separable over C and C ⊂ the center of V_{A}(D) = D ⊂ F = the center of F, F is separable over D. Thus, F is splitting ring for V_{A}(D) over D.

( ⟸ ) Let F be splitting ring for V_{A}(D) over D. Then D ⊂ F and F = V_{V_{A}(D)}(F), and so V_{A}(F) ⊂ V_{A}(D). Hence V_{A}(F) = V_{A}(D)∩V_{A}(F) = V_{V_{A}(D)}(F) = F. Thus F is a maximal commutative subalgebra of A. Moreover, since F is separable over D and D is separable over C, F is separable over C. Therefore, F is splitting ring for A.

Theorem 4.2. Assume B is a DeMeyer-Kanzaki G-Galois extension (that is, B is an Azumaya C-algebra and C is a G-Galois extension of C^{G} with G|_{C} ≅ G). Then, F is a splitting ring for the Azumaya algebra B ∗ G containing C if and only if F is a splitting ring for the Azumaya algebra B.

Proof. ( ⟹ ) Assume F is a splitting ring for the Azumaya algebra B ∗ G containing C. Then C ⊆ F and F = V_{B∗G}(F). Hence F = V_{B∗G}(F) ⊆ V_{B∗G}(G). Since C is a G-Galois extension of C^{G}, V_{B∗G}(C) = B by Lemma 3.2-(d). Thus V_{B∗G}(F) ⊆ V_{B∗G}(C) = B. Therefore V_{B∗G}(F) = V_{B}(F). But then F = V_{B∗G}(F) = V_{B}(F); and so F is a splitting ring for B.

( ⟸ ) Let F be a splitting ring for the Azumaya algebra B. Then C ⊆ F and F = V_{B}(F). Hence V_{B∗G}(F) ⊆ V_{B∗G}(C). By Lemma 3.2-(d) again, V_{B∗G}(C) = B, so V_{B∗G}(F) ⊆ V_{B∗G}(C) = B. Thus V_{B∗G}(F) = V_{B}(F); and so F = V_{B}(F) = V_{B∗G}(F). Therefore, F is a splitting ring for the Azumaya algebra B ∗ G containing C.

Next, we consider another Galois condition on B.

Theorem 4.3. Let B be a G-Galois extension of B^{G}, n^{−1} ∈ B and B ∗ G an Azumaya algebra. Then, F is a splitting ring for B ∗ G containing D, where D is the center of (B ∗ G)\overline{G} if and only if F is a splitting ring for V_{B∗G}(D).

Proof. This is an immediate consequence of Theorem 4.1-(ii) for the Azumaya algebra B ∗ G.

Corollary 4.4. Assume B is a G-Galois extension of B^{G}, n^{−1} ∈ B and B ∗ G an Azumaya algebra. Let G be an Abelian group. Then, F is a splitting ring for B ∗ G containing ZG if and only if F is a splitting ring for (B ∗ G)\overline{G}.

Proof. Since G is Abelian, n^{−1} ∈ B and Z is the center of B ∗ G, ZG is a commutative separable subalgebra. Let D = ZG. Then D is the center of (B ∗G)\overline{G} by Theorem 3.4. Moreover, V_{B∗G}(D) = V_{B∗G}(ZG) = (B ∗G)\overline{G}, so by Theorem 4.3, F is a splitting ring for B ∗ G containing ZG (= D) if and only if F is a splitting ring for (B ∗ G)\overline{G} (= V_{B∗G}(D)).

Theorem 4.5. Assume B is a G-Galois extension of B^{G}, n^{−1} ∈ B and B ∗ G is Azumaya algebra. Let F be a splitting ring for B ∗ G containing D, where D is the center of (B ∗ G)\overline{G}. Then, F is a splitting ring for (B ∗ G)\overline{G} if and only if G is Abelian.

Proof. ( ⟹ ) Since F is a splitting ring for B ∗ G, F = V_{B∗G}(F). Now, F = V_{(B∗G)\overline{G}}(F), so F = V_{(B∗G)\overline{G}}(F) = (B ∗ G)\overline{G} ∩ V_{B∗G}(F) = (B ∗ G)\overline{G} ∩ F. Thus

F ⊂ (B ∗ G)\overline{G}, and so F ⊂ V_{B∗G}(ZG). Therefore, V_{B∗G}(V_{B∗G}(ZG)) ⊂ V_{B∗G}(F) = F. Since n^{−1} ∈ B, ZG is a separable Z-algebra. Hence V_{B∗G}(V_{B∗G}(ZG)) = ZG because B ∗ G is an Azumaya Z-algebra ([4], Theorem 4.3, p. 57). Thus, ZG ⊂ F. But F is commutative, so G is Abelian.

( ⟸ ) Assume G is Abelian. Since Z is the center of B∗G, ZG is commutative. Hence ZG ⊂ F, and so F = V_{B∗G}(F) ⊂ V_{B∗G}(ZG). Thus F = V_{B∗G}(F) =

68 G. Szeto, L. Xue

V_{B∗G}(ZG) ∩ V_{B∗G}(F) = (B ∗ G)\overline{G} ∩ V_{B∗G}(F) = V_{(B∗G)\overline{G}}(F). Therefore, F is a splitting ring for (B ∗ G)\overline{G}.

By Corollary 4.4 and Theorem 4.5, under the hypothesis of Theorem 4.3, two of the following statements imply the third:

(1) F is a splitting ring for B ∗ G containing the center of (B ∗ G)\overline{G}.

(2) F is a splitting ring for (B ∗ G)\overline{G}.

(3) G is Abelian. We conclude the present paper with two examples of skew group rings B ∗ G

to show the relationship of the splitting rings between B ∗ G, B and (B ∗ G)\overline{G}.

Example 1. Let B = Q[i, j, k] = Q + Qi + Qj + Qk be the quaternion algebra over the rational field Q, G = { g_{1} = 1, g_{i}, g_{j}, g_{k} | g_{i}(x) = ixi^{−1}, g_{j}(x) = jxj^{−1}, g_{k}(x) = kxk^{−1} for all x ∈ B }, and A = B ∗ G. Then

(1) B is a G-Galois extension of B^{G} with G-Galois system \{\frac{1}{2}, −\frac{1}{2}i, −\frac{1}{2}j, −\frac{1}{2}k; \frac{1}{2}, \frac{1}{2}i, \frac{1}{2}j, \frac{1}{2}k\} and 4^{−1} ∈ B.

(2) B^{G} = Q, so A is an Azumaya Q-algebra ([1], Theorem 3.1).

(3) D = Q[i] = Q + Qi is a commutative separable Q-subalgebra of A.

(4) V_{A}(D) = D + Dg_{i} + (Qj + Qk)g_{j} + (Qj + Qk)g_{k} is an Azumaya D-algebra by Theorem 4.1-(i).

(5) F = D + Dg_{i} is a splitting ring for V_{A}(D), so, by Theorem 4.1-(ii), F = D + Dg_{i} is also a splitting ring for A.

(6) (B∗G)\overline{G} = V_{B∗G}(QG) = QG which is a commutative separable subalgebra, so QG is a splitting ring for (B ∗ G)\overline{G} (= QG) and for B ∗ G by Theorem 4.3 (or Corollary 4.4 for G is Abelian).

Example 2. Let M_{2}(Q) be the matrix ring of order 2 over the rational field Q, B = M_{2}(Q) ⊕ M_{2}(Q), g: B → B by g(a, b) = (b, a) for all (a, b) ∈ B. Then,

(1) g is an automorphism of B of order 2.

(2) Let G = {1, g}. Then B is a G-Galois extension of B^{G} with the Galois system {a_{1} = (I, 0), a_{2} = (0, I); b_{1} = (I, 0), b_{2} = (0, I)}, that is, a_{1}b_{1} + a_{2}b_{2} = (I, I) and a_{1}g(b_{1}) + a_{2}g(b_{2}) = (0, 0), where I is the identity of M_{2}(Q) and 0 is the zero matrix in M_{2}(Q).

(3) Let C be the center of B. Then C = Q ⊕ Q, and C is a G-Galois extension of C^{G} with the same Galois system as B and C|_{G} ≅ G.

(4) B ∗ G is an Azumaya C^{G}-algebra where C^{G} = { (a, a) | a ∈ Q } since B is an Azumaya C-algebra by Theorem 3.2.

(5) (B ∗ G)\overline{G} = C^{G} + C^{G}g.

(6) Since C is a commutative separable subalgebra of B ∗ G, V_{B∗G}(C) is an Azumaya C-algebra by Theorem 4.1-(i).

(7) V_{B∗G}(C) = B by Lemma 3.1-(d).

(8) Let F = Q [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] + Q [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ]. Then F is a separable maximal commutative subalgebra of M_{2}(Q), and so F ⊕F is a separable maximal commutative subalgebra of B, that is, F ⊕ F is a splitting ring for B. Thus, F ⊕ F is a splitting ring for B ∗ G by Theorem 4.2.

REFERENCES

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[3] Alfaro, R. and Szeto, G., The centralizer on H-separable skew group rings, Rings, Extension and Cohomology, Vol. 159, 1995.

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(received 08.11.1999.)

Department of Mathematics, Bradley University, Peoria, Illinois 61625, USA