S-r-ideals in commutative rings

Résumé

The rings considered in this article are commutative with identity. This article is motivated by the results  proved by Visweswaran (\cite{S-primary}) on $S$-primary ideals. In this paper, we introduce the concept of $S$-$r$-ideal (resp., $S$-$pr$-ideal) of a commutative ring and study its properties. Let $R$ be a commutative ring with $1\neq0$ and $S$ be a multiplicatively closed subset of $R$. Let $I$ be an ideal of $R$  disjoint with $S.$ We say that $I$ is an $S$-{\it r}-ideal (resp., $S$-{\it pr}-ideal) of $R$ if there exists an $s \in S$ such that for all $a, b \in R$ if $ ab \in I$ with $Ann(a) = (0)$ implies that  $sb\in I$ (resp.,  $sb \in \sqrt{I}$). We investigate many properties and characterizations of $S$-$r$-ideals (resp., $S$-$pr$-ideals). We discuss the form of $S$-$r$-ideals (resp., $S$-$pr$-ideals) in a finite direct product of rings. Furthermore, we study $S$-$r$-ideals (resp., $S$-$pr$-ideals) in Nagata's idealization ring. Our results allow us to construct original examples of $S$-$r$-ideals (resp., $S$-$pr$-ideals).