S-n-ideals of Commutative Rings

TL;DR

This paper introduces and studies $S$-$n$-ideals in commutative rings, generalizing $n$-ideals, and explores their properties, relationships, and specific cases including direct products, localizations, and special ring constructions.

Contribution

It defines $S$-$n$-ideals, investigates their properties, relationships with other ideals, and provides complete classifications in certain rings and algebraic constructions.

Findings

$S$-$n$-ideals generalize $n$-ideals and relate to $S$-prime and $S$-primary ideals.

Characterizations of $S$-$n$-ideals under various ring constructions are provided.

Complete determination of $S$-$n$-ideals in $ ext{Z}_m$ rings for specific $S$.

Abstract

Let $R$ be a commutative ring with identity and $S$ a multiplicatively closed subset of $R$. This paper aims to introduce the concept of $S$-$n$-ideals as a generalization of $n$-ideals. An ideal $I$ of $R$ disjoint with $S$ is called an $S$-$n$-ideal if there exists $s\in S$ such that whenever $ab\in I$ for $a,b\in R$, then $sa\in\sqrt{0}$ or $sb\in I$. The relationship among $S$% -$n$-ideals, $n$-ideals, $S$-prime and $S$-primary ideals are clarified. Several properties, characterizations and examples are presented such as investigating $S$-$n$-ideals under various contexts of constructions including direct products, localizations and homomorphic images. Furthermore, for $m\in% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $ and some particular $S$, all $S$-$n$-ideals of the ring $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{m}$ are completely determined. The last section is devoted for studying the $S$-$n$-ideals of the idealization ring and amalgamated algebra.