Two Generalizations of the Wedderburn-Artin Theorem with Applications

Abstract

We say that an $R$-module $M$ is {\it virtually simple} if $M\ne (0)$ 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 $k,n_1,\dots ,n_k\in \mathbb{N}$ 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 $R{m}_1\oplus \dots \oplus R{m}_k$ where each $R{m}_i$ is either a simple $R$-module or a left virtually simple direct summand of $R$.

18 Pages. arXiv admin note: text overlap with arXiv:1603.05647