Partial actions of groups on generalized matrix rings

Abstract

Let $n$ be a positive integer and $R=(M_{ij})_{1\leq i,j\leq n}$ be a generalized matrix ring. For each $1\leq i,j\leq n$, let $I_i$ be an ideal of the ring $R_i:=M_{ii}$ and denote $I_{ij}=I_iM_{ij}+M_{ij}I_j$. We give sufficient conditions for the subset $I=(I_{ij})_{1\leq i,j\leq n}$ 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 $1\leq i\leq n$. 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.