Seminoetherian Modules over Non-Primitive HNP rings

TL;DR

The paper addresses the problem of classifying seminoetherian right modules over non-primitive hereditary noetherian prime (HNP) rings. It develops a structural framework centered on the largest singular submodule $T$ of a module $M$, a finite-dimensional non-singular quotient $M/T$, and an essential submodule $X$ with $T subseteq X$, such that each primary component of $T$ and of $M/X$ splits into direct sums of cyclic uniserial modules with bounded total length. The main achievement is Theorem 1.1, which provides equivalent conditions for seminoetherianity and yields a detailed description of the componentwise behavior of singular parts, along with corollaries and explicit examples illustrating max modules and Bezout/distributive phenomena. The results extend parallels with torsion theories in abelian groups and provide a concrete decomposition framework for seminoetherian modules over HNP rings, including implications for injective hulls and primary decomposition. Together, these findings give a comprehensive, constructive picture of when and how seminoetherian modules over non-primitive HNP rings decompose into well-understood cyclic uniserial building blocks, with practical guidance for identifying and constructing such modules.

Abstract

We study the structure of seminoetherian modules. Seminoetherian modules over non-primitive hereditary noetherian prime rings are completely described.