Partial actions of groups on generalized matrix rings

Abstract

Let $n$ be a positive integer and $R=(M_{ij}{)}_{1\le i,j\le n}$ be a generalized matrix ring. For each $1\le i,j\le n$, let $I_i$ be an ideal of the ring $R_i:=M_{ii}$ and denote $I_{ij}=I_i{M}_{ij}+M_{ij}{I}_j$. We give sufficient conditions for the subset $I=(I_{ij}{)}_{1\le i,j\le n}$ of $R$ to be an ideal of $R$. Also, suppose that ${\alpha }^{(i)}$ is a partial action of a group $\mathtt{G}$ on $R_i$, for all $1\le i\le n$. We construct, under certain conditions, a partial action $\gamma$ of $\mathtt{G}$ on $R$ such that $\gamma$ restricted to $R_i$ coincides with ${\alpha }^{(i)}$. 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 $R^{\gamma }\subset R$. Some examples to illustrate the results are considered in the last part of the paper.