The Generalized 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 Ii = {c - gi(c) | c ∈ C} for each gi ∈ G. Then, B is called a center Galois extension with Galois group G if BIi = B for each gi ≠ 1 in G, and a weak center Galois extension with group G if BIi = Bei for some nonzero idempotent ei in C for each gi ≠ 1 in G. When ei is a minimal element in the Boolean algebra generated by {ei | gi ∈ G} Bei is a center Galois extension with Galois group Hi for some subgroup Hi of G. Moreover, the central Galois algebra B(1 – ei) is characterized when B is a Galois algebra with Galois group G.