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

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. 