The structure of piecewise prime modules

摘要

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. Communicated by Toma Albu