Virtually uniserial modules and rings

Abstract

We study the class of virtually uniserial modules and rings as a nontrivial generalization of uniserial modules and rings. An R-module M is virtually uniserial if for every finitely generated submodule 0≠K⊆M, K/Rad(K) is virtually simple. Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules and a left virtually uniserial (resp., left virtually serial) ring is a ring which is virtually uniserial (resp., serial) as a left R-module. We give some useful properties of virtually (uni)serial modules and rings. In particular, it is shown that every left virtually uniserial module is uniform and Bézout. Also, we show that if R is a left virtually serial ring, then R/J(R)≅∏i=1tMni(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain. As a consequence, we obtain that a ring R is left virtually serial with J(R)=0 if and only if R≅∏i=1tMni(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain with J(Di)=0. Also, several classes of rings for which every virtually uniserial module (resp., ring) is uniserial are given. Noetherian left virtually uniserial rings are characterized. Finally, we obtain some structure theorems for (commutative) rings over which every (finitely generated) module is virtually serial.