Virtually Semisimple Modules and Two Generalizations of Wedderburn-Artin and Krull-Schmidt Theorems with Applications

An $R$-module $M$ is called {\it virtually semisimple} if each submodule of $M$ is isomorphic to a direct summand of $M$, and $M$ is called {\it virtually simple} if $M\neq (0)$ and $N\cong M$ for each $0\neq N\leq M$. By using these notions, we give natural generalizations of the Wedderburn-Artin and Krull-Schmidt Theorems. 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) $R$-module is virtually semisimple; (ii) Every finitely generated left (right) $R$-module is a direct sum of virtually simple modules; (iii) $R\cong\prod_{i=1}^{k} M_{n_i}(D_i)$ for some principal ideal V-domains $D_i$ s; and {\rm (iv)} Every non-zero finitely generated left $R$-module can be written uniquely in the form $ Rm_1 \oplus\ldots\oplus Rm_k$ where each $Rm_i$ is a simple $R$-module or a left virtually simple direct summand of $R$.