The structure of piecewise prime modules

Abstract

The purpose of this paper is to further the study of quasi-Baer modules by investigating the structure of a special class of quasi-Baer modules. As a module theoretic analog of a piecewise prime ring, we define and characterize a piecewise prime module (simply, a PWP module) via its endomorphism ring. Although it is well known that eRe is a PWP ring for a left (right) semicentral idempotent or a full idempotent e in a PWP ring R, it is still an open question whether this holds true when e is an arbitrary idempotent. We give an affirmative answer to this question. As a consequence, we prove that every direct summand of a PWP module is a PWP module. It is shown that any direct sum of copies of a PWP module is always a PWP module. Consequently, every column (and row) finite matrix ring over a PWP ring is a PWP ring. We obtain a complete structure theorem for PWP modules and show that endoprime submodules are the building blocks of the PWP modules. Applications and examples illustrating our results are provided.