George Szeto and Lianyong Xue

Copyright © 2001 George Szeto and Lianyong Xue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.