On Rings over Which All Modules Are Direct Sums of Distributive Modules

TL;DR

This paper investigates rings over which every right module decomposes into a direct sum of distributive modules, exploring the structural properties that ensure all modules are semidistributive.

Contribution

It characterizes rings where all modules are semidistributive, advancing understanding of module decompositions and lattice distributivity over rings.

Findings

Identifies conditions for rings with all modules semidistributive

Provides classification results for such rings

Connects module lattice properties to ring structure

Abstract

A module is said to be \textsf{distributive} if the lattice of its submodules is distributive. A direct sum of distributive modules is called a \textsf{semidistributive} module. In this paper we consider rings $A$ such that all right $A$-modules are semidistributive.