The Galois algebras and the Azumay Galois extensions

Abstract

Let B be a Galois algebra over a commutative ring R with Galois group G, C the center of B, K={g∈G|g(c)=c for all  c∈C}, Jg{b∈B|bx=g(x)b for all x∈B} for each g∈K, and BK=(⊕∑g∈K Jg). Then BK is a central weakly Galois algebra with Galois group induced by K. Moreover, an Azumaya Galois extension B with Galois group K is characterized by using BK.