Two Generalizations of the Wedderburn-Artin Theorem with Applications

TL;DR

This paper introduces two new generalizations of the Wedderburn-Artin Theorem using virtually simple and virtually semisimple modules, providing structural characterizations of rings where finitely generated modules decompose into these types.

Contribution

It develops a theoretical framework extending classical theorems to virtually simple modules, with applications to ring decompositions and module structures.

Findings

Characterization of rings where finitely generated modules are virtually semisimple

Equivalence of ring decompositions with products of matrix rings over principal ideal V-domains

Unique decomposition of finitely generated modules into simple or virtually simple summands

Abstract

We say that an $R$-module $M$ is {\it virtually simple} if $M\neq (0)$ and $N\cong M$ for every non-zero submodule $N$ of $M$, and {\it virtually semisimple} if each submodule of $M$ is isomorphic to a direct summand of $M$. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules. Our theory provides two natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring $R$: (i) Every finitely generated left (right) $R$-modules is virtually semisimple; (ii) Every finitely generated left (right) $R$-modules is a direct sum of virtually simple modules; (iii) $R\cong\prod_{i=1}^{k} M_{n_i}(D_i)$ where $k, n_1,\ldots,n_k\in \Bbb{N}$ and each $D_i$ is a principal ideal V-domain; and {\rm (iv)} Every non-zero finitely generated left $R$-module can be written uniquely (up to isomorphism and order of the factors) in the form $ Rm_1 \oplus\ldots\oplus Rm_k$ where each $Rm_i$ is either a simple $R$-module or a left virtually simple direct summand of $R$.