Generalized Bassian Modules over Non-primitive Dedekind Prime Rings

TL;DR

The paper addresses the classification of generalized bassian modules over non-primitive Dedekind prime rings by analyzing singular modules with primary decompositions. It develops a suite of general results on closure, torsion theory, and Goldie dimension, and then provides a detailed structure theory for modules over non-primitive HNP rings. Central to the work is Theorem 1.2, which equates generalized bassianness with being a direct sum of a Bassian and a semisimple module and with each primary component splitting into a noetherian plus semisimple part. The findings yield a clear, componentwise description of generalized bassian singular modules and lay groundwork for extending these results to broader ring classes, with open questions about HNP rings remaining.

Abstract

A right $A$-module $M$ is said to be generalized bassian if the existence of an injective homomorphism $M\to M/N$ for some submodule $N$ of $M$ implies that $N$ is a direct summand of $M$. We describe singular generalized bassian modules over non-primitive Dedekind prime rings.\\ The study is supported by grant of Russian Science Foundation.