Hereditarily and Super Bassian Modules over Certain Rings

TL;DR

This paper characterizes hereditarily and super Bassian modules over certain rings, providing structural criteria and exploring their properties over Dedekind prime rings, including conditions under which modules are Bassian.

Contribution

It introduces new criteria for hereditarily and super Bassian modules over Dedekind prime rings, extending recent results and clarifying their relationships.

Findings

Singular modules over Dedekind prime rings are super Bassian iff they are Bassian.

Super Bassian modules over these rings are always hereditary Bassian.

The paper extends recent results by Tuganbaev and Blacher on Bassian modules.

Abstract

We characterize in certain basic cases when a module over a ring is either {\it hereditarily Bassian} or {\it super Bassian} in the sense that either each its proper submodule is Bassian or, respectively, each its proper epimorphic image is Bassian. We prove several structural criteria for both hereditarily Bassian and super Bassian modules over non-primitive Dedekind prime rings, and in particular Dedekind domains. Over these rings, we establish that a singular module is super Bassian exactly when it is Bassian, which is true if and only if it is Bassian. In addition, for an arbitrary (not necessarily singular) module over a non-primitive Dedekind prime ring, the property of being super Bassian curiously implies the property of being hereditary Bassian always. Our results somewhat continue and supply recent results due to Tuganbaev in Mathematics (2026) and Blacher in J. Algebra (2026).