On weak center Galois extensions of rings

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, the notion of a center Galois extension of B G with Galois group G (i.e., C is a Galois algebra over C G 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 B I i = B e i for some idempotent in C and I i = { c − g i ( c ) | c ∈ C } for each g i ≠ 1 in G . It is shown that B is a weak center Galois extension with group G if and only if for each g i ≠ 1 in G there exists an idempotent e i in C and { b k e i ∈ B e i ; c k e i ∈ C e i , k = 1 , 2 , ... , m } such that ∑ k = 1 m b k e i g i ( c k e i ) = δ 1 , g i e i and g i restricted to C ( 1 − e i ) is an identity, and a structure of a weak center Galois extension with group G is also given.