On partial Galois Azumaya extensions

Abstract

Let \(\alpha\) be a globalizable partial action of a finite group \(G\) over a unital ring \(R\), \(A=R\star_\alpha G\) the corresponding partial skew group ring, \(R^\alpha\) the subring of the \(\alpha\)-invariant elements of \(R\) and \(\alpha^\star\) the partial inner action of \(G\) (induced by \(\alpha\)) on the centralizer \(C_A(R)\) of \(R\) in \(A\). In this paper we present equivalent conditions to characterize \(R\) as an \(\alpha\)-partial Galois Azumaya extension of \(R^\alpha\) and \(C_A(R)\) as an \(\alpha^\star\)-partial Galois extension of the center \(C(A)\) of \(A\). In particular, we extend to the setting of partial group actions similar results due to R. Alfaro and G. Szeto [1,2,3].