S-1-absorbing primary submodules

TL;DR

This paper introduces the concept of S-1-absorbing primary submodules as an extension of existing notions, exploring their properties, characterizations, and applications in idealization and amalgamation within module theory.

Contribution

It defines S-1-absorbing primary submodules, investigates their properties, and provides new theorems including avoidance and applications to idealization and amalgamation.

Findings

Characterization of S-1-absorbing primary submodules

S-1-absorbing primary avoidance theorem

Applications to idealization and amalgamation

Abstract

In this work, we introduce the notion of $S$-1-absorbing primary submodule as an extension of 1-absorbing primary submodule. Let $S$ be a multiplicatively closed subset of a ring $R$ and $M$ be an $R$-module. A submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be $S$-1-absorbing primary if whenever $abm\in N$ for some non-unit $a,b\in R$ and $m\in M$, then either $sab\in(N:_{R}M)$ or $sm\in M$-$rad(N)$. We examine several properties of this concept and provide some characterizations. In addition, $S$-1-absorbing primary avoidance theorem and $S $-1-absorbing primary property for idealization and amalgamation are presented.