# Structure Theorem for Rings Whose Finitely Generated Modules are Direct
Sums of Virtually Simple Modules

Research paper by **Mahmood Behboodi, Asghar Daneshvar, Mohammad Reza Vedadi**

Indexed on: **29 Jun '16**Published on: **29 Jun '16**Published in: **Mathematics - Rings and Algebras**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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 $R$. Finally, we characterize finitely generated virtually semisimple
modules over commutative rings.