Partial actions of groups on generalized matrix rings

Abstract

Let n be a positive integer and R=(Mij)1≤i,j≤n be a generalized matrix ring. For each 1≤i,j≤n, let Ii be an ideal of the ring Ri:=Mii and denote Iij=IiMij+MijIj. We give sufficient conditions for the subset I=(Iij)1≤i,j≤n of R to be an ideal of R. Also, suppose that α(i) is a partial action of a group G on Ri, for all 1≤i≤n. We construct, under certain conditions, a partial action γ of G on R such that γ restricted to Ri coincides with α(i). We study the relation between this construction and the notion of Morita equivalent partial group action given in [1]. Moreover, we investigate properties related to Galois theory for the extension Rγ⊂R. Some examples to illustrate the results are considered in the last part of the paper.