Second Representable Modules over Commutative Rings

TL;DR

This paper studies second representable modules over commutative rings, providing conditions for their minimal presentations and exploring associated prime ideals, advancing understanding of module decomposition.

Contribution

It introduces the concept of second representable modules, offers criteria for minimal second presentations, especially in lifting modules, and analyzes related prime ideals.

Findings

Established sufficient conditions for second representability.

Characterized minimal second presentations in lifting modules.

Analyzed the structure of second attached prime ideals.

Abstract

Let $R$ be a commutative ring. We investigate $R$-modules which can be written as \emph{finite} sums of {\it {second}} $R$-submodules (we call them \emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be have a (minimal) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (\emph{main}) \emph{second attached prime ideals} related to a module with such a presentation.