On Weakly S-primary Submodules

TL;DR

This paper introduces and studies the concept of weakly S-primary submodules in modules over commutative rings, exploring their properties, characterizations, and behavior under various module operations.

Contribution

It defines weakly S-primary submodules, investigates their properties, and examines their behavior under homomorphisms, localizations, quotients, and other module constructions.

Findings

Characterization of weakly S-primary submodules in finitely generated faithful multiplication modules

Behavior of weakly S-primary submodules under module homomorphisms and localizations

Conditions for submodules of amalgamation modules to be weakly S-primary

Abstract

Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ to be weakly $S$-primary if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in\sqrt{(N:_{R}M)}$ or $sm\in N$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Moreover, the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations is investigated. Finally, we determine some conditions under which two kinds of submodules of the amalgamation module along an ideal are weakly $S$-primary.