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

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 ${}_{R}M$ 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 ${}_{R}R$ 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 $k,n_1,...,n_k\in \mathbb{N}$ 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 $R$.