© Hindawi Publishing Corp. 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 ∑m k=1bkeigi(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. 2000 Mathematics Subject Classification. Primary 16S35, 16W20. 1. Introduction. Galoi