On central commutator Galois extensions of rings

Abstract

Abstract. Let B be a ring with 1, G a finite automorphism group of B of order n for some integer n, B G the set of elements in B fixed under each element in G, and ∆ = V B (B G

On a Class of Azumaya Galois Extensions

“…Proof. Since B is a commutator Galois extension of B G with Galois group G, V B (B G ) is a Galois algebra over C G with Galois group G by Lemma 3.1 in [8]. But then B and…”

“…A Galois extension B with Galois group G is called an Azumaya Galois extension if B G is an Azumaya algebra over C G ([2], [6]), and a DeMeyer-Kanzaki Galois extension if B is an Azumaya algebra over C which is a Galois algebra over C G with Galois group induced by and isomorphic with G ([3], [5]). A Galois extension B with Galois group G is called a commutator Galois extension of B G if the commutator subring of B G in B, V B (B G ), is a Galois extension with Galois group induced by and isomorphic with G ( [8]). We note that the class of DeMeyer-Kanzaki Galois extensions is contained in the class of Azumaya Galois extensions which is a subclass of commutator Galois extensions.…”

“…In [5], the result was generalized to a Galois algebra B with nontrivial idempotents: B is a central Galois algebra with Galois group K and C is a commutative Galois algebra with Galois group G=K if and only if J g = f0g for each g T P K where J g = fb P B j bx = g(x)b for all x P Bg for an g P G. The purpose of the present paper is to give a further generalization to the class of Azumaya Galois extensions for which J g = f0g for each g T P K where an Azumaya Galois extension B with Galois group G is a Galois extension of B G which is an Azumaya C G -algebra (for more about an Azumaya Galois extension, see [2] and [6]). We recall that a Galois extension B with Galois group G is a commutator Galois extension if the commutator subring of B G in B, V B (B G ), is a Galois extension with Galois group induced by and isomorphic with G ( [8]), and a DeMeyer-Kanzaki Galois extension if B is an Azumaya algebra over C which is a Galois algebra over C G with Galois group induced by and isomorphic with G ( [3], [5]). We shall show the following equivalent conditions:…”