The general Ikehata theorem forH-separable crossed products

George Szeto,Lianyong Xue

doi:10.1155/s0161171200003124

George Szeto, Lianyong Xue

Open Access

https://doi.org/10.1155/s0161171200003124

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Abstract

LetBbe a ring with1, Cthe center ofB, Gan automorphism group ofBof ordernfor some integern, CGthe set of elements inCfixed underG, Δ=Δ(B,G,f)a crossed product overBwherefis a factor set fromG×GtoU(CG). It is shown thatΔis anH-separable extension ofBandVΔ(B)is a commutative subring ofΔif and only ifCis a Galois algebra overCGwith Galois groupG|C≅G.

Highlights

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, x2, . . . , xn−1} and xn = v ∈ U (Bρ) for some integer n, where Bρ is the set of elements in B fixed under ρ and U (Bρ) is the set of units of Bρ.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 Cρ with Galois group ρ|C generated by ρ|C of order n
The purpose of the present paper is to generalize the above Ikehata theorem to an automorphism group of B and f is an factor set from G × G to U (CG)
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 CG with Galois group G|C G

Summary

Introduction

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 Cρ with Galois group ρ|C generated by ρ|C of order n. 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 CG with Galois group G|C G.

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