$\aleph_0$-distributive modules and rings

TL;DR

This paper characterizes rings and modules with specific distributive properties, showing their equivalence to well-known classes like Artinian, Noetherian, and finite representation type rings, advancing the understanding of module and ring structures.

Contribution

It establishes new equivalences between $eth_0$-distributive modules/rings and classical classes such as Artinian, Noetherian, and finite representation type rings.

Findings

$eth_0$-distributive modules coincide with Artinian/Noetherian modules.

Rings with all modules as $eth_0$-distributive are of finite representation type.

Rings with semidistributive modules are Kawada rings.

Abstract

Let $A$ be a ring with minimum condition on principal right ideals. It is proved that $\aleph_0$-distributive right (left) $A$-modules coincide with Artinian (Noetherian) right (left) $A$-modules. Rings, over which all right modules are direct sums of $\aleph_0$-distributive coincide with rings of of finite representation type. Rings, whose right modules are semidistributive, coincide with Kawada rings, over basis rings of which all right modules are completely cyclic. The studies of Tuganbaev are supported by Russian Scientific Foundation, project 22-11-00052.