On characterizations of a center Galois extension

Abstract

Abstract. Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and B G the set of elements in B fixed under each element in G. Then, it is shown that B is a center Galois extension of B G (that is, C is a Galois algebra over C G with Galois group G| C G) if and only if the ideal of B generated by{c − g(c) | c ∈ C} is B for each g ≠ 1 in G. This generalizes the well known characterization of a commutative Galois extension C that C is a Galois extension of C G with Galois group G if and only if the ideal generated by {c − g(c) | c ∈ C} is C for each g ≠ 1 in G. Some more characterizations of a center Galois extension B are also given.

On a Class of Azumaya Galois Extensions

“…By using the similar argument as given in Theorem 3.5, we can show two equivalent conditions for a commutator Galois extension for which J g = f0g for each g T P K. Recall that B is called a center Galois extension of B G with Galois group G if C is a Galois algebra over C G with Galois group G ( [7]).…”

“…As given in [7], B is called a Galois extension of B G with Galois group G if there exist elements fa i ; b i in B, i = 1; 2; :::; m for some integer mg such that m i=1 a i g(b i ) = 1;g , and such a set fa i ; b i g is called a G-Galois system for B. 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]).…”

On Azumaya algebras with a finite automorphism group

“…We denote V B (A) the commutator subring of A in B. We follow the definitions of a Galois extension, a separable extension, and an Azumaya algebra as given in [1,5,7]. The ring B is called 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 3.…”