Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem

Abstract

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them virtually semisimple modules. A module R M is called completely virtually semisimple if each submodules of M is a virtually semisimple module. A ring R is then called left (completely) virtually semisimple if R R is a left (compleatly) virtually semisimple R-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring R is left completely virtually semisimple if and only ifwhere k, n 1 , ..., n k ∈ N and each D i is a principal left ideal domain. Moreover, the integers k, n 1 , ..., n k and the principal left ideal domains D 1 , ..., D k are uniquely determined (up to isomorphism) by R. * The research of the first author was in part supported by a grant from IPM (No. 94130413). This research is partially carried out in the IPM-Isfahan Branch.†

Several Generalizations of the Wedderburn-Artin Theorem with Applications

“…We are now in a position to give the following generalization of the Wedderburn-Arttin Theorem. We remark that the equivalences between (2) and (3) below has been shown in [5,Theorem 3.13].…”

“…If each submodule of M is a virtually semisimple module, we call M completely virtually semisimple. We also have the following implications for R M : M is semisimple ⇒ M is completely virtually semisimple ⇒ M is virtually semisimple These implications are also irreversible in general (see [5,Examples 3.7 and 3.8]).…”

“…Being a left virtually semisimple ring is not Morita invariant (see [5,Example 3.8]). Surprisingly, being a left completely virtually semisimple ring is a Morita invariant property (see [5,Proposition 3.3]).…”

“…Being a left virtually semisimple ring is not Morita invariant (see [5,Example 3.8]). Surprisingly, being a left completely virtually semisimple ring is a Morita invariant property (see [5,Proposition 3.3]). In addition, being (completely) virtually semisimple module is a Morita invariant property (see [5, Proposition 2.1 (iv)]).…”

“…This property motivates us to study modules for which every submodule is isomorphic to a direct summand. In fact, the notions of "virtually semisimple modules" and "completely virtually semisimple modules" were introduced and studied in our recent work [5] as generalizations of semisimple modules. We recall that an R-module M is virtually semisimple if each submodule of M is isomorphic to a direct summand of M .…”