Virtually Semisimple Modules and a Generalization of the Wedderburn-Artin Theorem

TL;DR

This paper introduces the concept of virtually semisimple modules, generalizes the Wedderburn-Artin theorem, and characterizes rings that are completely virtually semisimple as products of matrix rings over principal left ideal domains.

Contribution

It defines virtually semisimple modules, extends the Wedderburn-Artin theorem, and characterizes rings with this property as specific matrix ring products over principal left ideal domains.

Findings

Characterization of left completely virtually semisimple rings as matrix rings over principal left ideal domains.

Introduction of virtually semisimple modules and their properties.

Uniqueness of the decomposition parameters for such rings.

Abstract

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them {\it virtually semisimple modules}. A module $_RM$ is called {\it completely virtually semisimple} if each submodules of $M$ is a virtually semisimple module. A ring $R$ is then called {\it left} ({\it completely}) {\it virtually semisimple} if $_RR$ is a left (compleatly) virtually semisimple $R$-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring $R$ is left completely virtually semisimple if and only if $R \cong \prod _{i=1}^ k M_{n_i}(D_i)$ where $k, n_1, ...,n_k\in \Bbb{N}$ and each $D_i$ is a principal left ideal domain. Moreover, the integers $k,~ n_1, ...,n_k$ and the principal left ideal domains $D_1, ...,D_k$ are uniquely determined (up to isomorphism) by $R$.