Several Generalizations of the Wedderburn-Artin Theorem with Applications

Abstract

We say that an R-module M is virtually semisimple if each submodule of M is isomorphic to a direct summand of M. A nonzero indecomposable virtually semisimple module is then called a virtually simple module. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules . Our study provides several 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-module is virtually semisimple; (ii) Every finitely generated left (right) R-module is a direct sum of virtually simple R-modules; (iii) $R\cong {\prod }_{i = 1}^{k} M_{n_{i}}(D_{i})$ where k,n ₁,…,n _k ∈ ℕ and each D _i is a principal ideal V-domain; and (iv) Every nonzero finitely generated left R-module can be written uniquely (up to isomorphism and order of the factors) in the form R m ₁ ⊕… ⊕ R m _k where each R m _i is either a simple R-module or a virtually simple direct summand of R.