PS-Hollow Representations of Modules over Commutative Rings

TL;DR

This paper introduces PS-hollow submodules in modules over commutative rings and studies their properties, especially focusing on modules with finite sums of such submodules and their minimal representations over Artinian rings.

Contribution

It defines PS-hollow submodules and establishes existence and uniqueness results for minimal PS-hollow strongly representations over Artinian rings.

Findings

Existence of minimal PS-hollow strongly representations.

Uniqueness of minimal PS-hollow strongly representations.

Application to modules over Artinian rings.

Abstract

Let $R$ be a commutative ring and $M$ a non-zero $R$-module. We introduce the class of \emph{pseudo strongly hollow submodules} (\emph{PS-hollow submodules}, for short) of $M$. Inspired by the theory of modules with \emph{secondary representations}, we investigate modules which can be written as \emph{finite} sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of \emph{minimal} PS-hollow strongly representations of modules over Artinian rings.