Direct sum decompositions of projective and injective modules into virtually uniserial modules

A theorem due to Warfield states that “a ring R is left serial if and only if every (finitely generated) projective left R-module is serial” and a theorem due to Tuganbaev states that “a ring R is a finite direct product of uniserial Noetherian rings if and only if R is left duo, and all injective left R-modules are serial”. Most recently, in our previous paper [Virtually uniserial modules and rings, J Algebra 549:365–385, 2020], we introduced and studied the concept of virtually uniserial modules as a nontrivial generalization of uniserial modules. We say that an R-module M is virtually uniserial if, for every finitely generated submodule 0≠K⊆M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\ne K\subseteq M$$\end{document}, K/Rad(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K/\mathrm{Rad}(K)$$\end{document} is virtually simple (an R-module M is virtually simple if, M≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\ne 0$$\end{document} and M≅N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\cong N$$\end{document} for every nonzero submodule N of M). Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules. The above results of Warfield and Tuganbaev motivated us to study the following questions: “Which rings have the property that every projective module is virtually serial?” and “Which rings have the property that every injective module is virtually serial?”. The goal of this paper is to answer these questions.