ON S-WEAKLY QUASI n-ABSORBING SUBMODULES AND THEIR EXTENSIONS

S-WEAKLY QUASI n-ABSORBING SUBMODULES

Résumé

Let $R$ be a commutative ring with identity, $S$ a multiplicative subset of $R$ and $n$ a positive integer. This paper aims to introduce the concept of $S$-weakly quasi $n$-absorbing submodules as a natural generalization of weakly quasi $n$-absorbing submodules. Specifically, a submodule $N$ of an $R$-module $M$ with $\left(N:_{R} M\right)\cap S=\emptyset$ is termed $S$-weakly quasi $n$-absorbing submodule if there exists an (fixed) $s\in S$ such that for some $a\in R$ and $m\in M$, whenever $0 \neq a^{n}m\in N$, it holds that either $sa^{n}\in\left(N:_{R} M\right)$ or $sa^{n-1}m \in N$. This element $s\in S$ is referred to as an $S$-weakly element of $N$. In addition to establishing various properties, characterizations, and illustrative examples of the concept, this study examines the behavior of $S$-weakly quasi $n$-absorbing submodules within a variety of algebraic framework, including localizations, homomorphic images, idealizations, and amalgamations.