On weak center Galois extensions of rings

Abstract

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and BG the set of elements in B fixed under each element in G. Then, the notion of a center Galois extension of BG with Galois group G (i.e., C is a Galois algebra over CG with Galois group G|C≅G) is generalized to a weak center Galois extension with group G, where B is called a weak center Galois extension with group G if BIi=Bei for some idempotent in C and Ii={c−gi(c)|c∈C} for each gi≠1 in G. It is shown that B is a weak center Galois extension with group G if and only if for each gi≠1 in G there exists an idempotent ei in C and {bkei∈Bei;ckei∈Cei,k=1,2,...,m} such that ∑k=1mbkeigi(ckei)=δ1,giei and gi restricted to C(1−ei) is an identity, and a structure of a weak center Galois extension with group G is also given.