THE GENERAL IKEHATA THEOREM FOR H-SEPARABLE CROSSED PRODUCTS

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, CG the set of elements in C fixed under G, ∆ = ∆(B,G,f ) a crossed product over B where f is a factor set from G×G to U(CG). It is shown that ∆ is an H-separable extension of B and V∆(B) is a commutative subring of ∆ if and only if C is a Galois algebra over CG with Galois group G|C G.


Introduction.
Let B be a ring with 1, ρ an automorphism of B of order n, B[x; ρ] a skew polynomial ring with a basis{1,x,x 2 ,...,x n−1 } and x n = v ∈ U(B ρ ) for some integer n, where B ρ is the set of elements in B fixed under ρ and U(B ρ ) is the set of units of B ρ .
In [4] it was shown that any skew polynomial ring B[x; ρ] of prime degree n is an H-separable extension of B if and only if C is a Galois algebra over C ρ with Galois group ρ| C generated by ρ| C of order n. This theorem was extended to any degree n [5, Theorem 1]. Recently, the theorem was completely generalized by the present authors in [8], that is, let B[x; ρ] be a skew polynomial ring of degree n for some integer n. Then, B[x; ρ] is an H-separable extension of B if and only if C is a Galois algebra over C ρ with Galois group ρ| C ρ . The purpose of the present paper is to generalize the above Ikehata theorem to an automorphism group of B (not necessarily cyclic) and f is an factor set from G × G to U(C G ). We show that ∆ is an H-separable extension of B and V ∆ (B) is a commutative subring of ∆ if and only if C is a Galois algebra over C G with Galois group G| C G.

Preliminaries and basic definitions.
Throughout this paper, B represents a ring with 1, C the center of B, G an automorphism group of B of order n for some integer n, B G the set of elements in B fixed under G, ∆ = ∆(B,G,f ) a crossed product with a free basis {U g | g ∈ G and U 1 = 1} over B and the multiplications are given by , Z the center of ∆,Ḡ the inner automorphism group of ∆ induced by G, that is,ḡ(x) = U g xU −1 g for each x ∈ ∆ and g ∈ G. We note that f (g,1) = f (1,g) = f (1, 1) = 1 for all g ∈ G andḠ restricted to B is G.
Let A be a subring of a ring S with the same identity 1. We denote V s (A) the commutator subring of A in S. A ring S is called a G-Galois extension of S G if there exist elements {a i ,b i ∈ S, i = 1, 2,...,m} for some integer m such that

The Ikehata theorem.
In this section, we show that ∆ is an H-separable extension of B and V ∆ (B) is a commutative subring of ∆ if and only if C is a Galois algebra over C G with Galois group G| C G. We begin with a lemma.
Since ∆ is an H-separable extension of B again, there exists an H-separable system 1, 2,...,m} is a G-Galois system for C. Therefore, C is a Galois algebra over C G with Galois group G | C G.
(⇐ ) Since C is a Galois algebra over C G with Galois group with G| C G, there exists a G-Galois system {a i ,b i ∈ C | i = 1, 2,...,m} for some integer m such that m i=1 a i g(b i ) = δ 1,g . Let x i = a i and y i = g∈G g(b i )U g ⊗ B U −1 g . We claim that {x i ∈ V ∆ (B), y i ∈ V ∆⊗ B ∆ (∆) | i = 1, 2,...,m} is an H-separable system for ∆ over B. In fact, (3.5) for any h ∈ G, Thus  Next we prove more characterizations of the ring B as given in Theorem 3.2.

Theorem 3.4. Assume ∆ is an H-separable extension of B. Then the following statements are equivalent:
(
(5) ⇒(1). Since C ⊂ B, J g ⊂ I g for all g ∈ G. Hence I g = {0} implies J g = {0}. But then V ∆ (B) = g∈G J g U g = J 1 = C which is commutative. We conclude the present paper with two examples of crossed products ∆ to demonstrate our results: (1) ∆ is an H-separable extension of B, but V ∆ (B) is not commutative, (2) V ∆ (B) is commutative, but ∆ is not an H-separable extension of B. Hence C is not a Galois algebra over C G with G | C G in either example by Theorem 3.2.
(2) ∆ is a separable extension of B and B is an Azumaya Q-algebra, so ∆ is an Azumaya Q-algebra. Since ∆ is a free left B-module, ∆ is an H-separable extension of B [3, Theorem 1].
(3) V ∆ (B) = Q + QiU g i + QjU g j + QkU g k which is not commutative, so C is not a Galois algebra over C G with Galois group G | C G by Theorem 3.2.

Then
(1) The center of B, C = Q = C G .
(3) The center of ∆, Z = Q + QiU g i ≠ C G . On the other hand, assume that ∆ is an H-separable extension of B. Since B is a direct summand of ∆ as a left B-module, V ∆ (V ∆ (B)) = B [7,Proposition 1.2]. This implies that the center of ∆, Z = C G , a contradiction. Thus ∆ is not an H-separable extension of B. Therefore, C is not a G-Galois algebra over C G with G| c G by Theorem 3.2.