Partial actions of groups on generalized matrix rings

TL;DR

This paper investigates partial group actions on generalized matrix rings, providing conditions for ideals, constructing compatible partial actions, and exploring their relation to Morita equivalence and Galois theory.

Contribution

It introduces a method to extend partial group actions from component rings to the entire generalized matrix ring and analyzes their properties and relations to existing concepts.

Findings

Provided sufficient conditions for ideals in generalized matrix rings.

Constructed partial actions on the entire ring from component actions.

Explored connections with Morita equivalence and Galois theory.

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.