S-n-ideals of Commutative Rings

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 $m\in$ and some particular $S$, all $S$-$n$-ideals of the ring $$ are completely determined. The last section is devoted for studying the $S$-$n$-ideals of the idealization ring and amalgamated algebra.