On a generalization of $r$-ideals in commutative rings

Abstract

Let $R$ be a commutative ring with nonzero identity, $J$ a proper ideal of $R$. In this paper, we present the concept of $H_J$-ideals in commutative rings. A proper ideal $I$ of R is called an $H_J$-ideal if whenever $a$, $b$ $\in R$ with $ab\in I$ and $(J:a) =J$ such that $(J:a)=\{x \in R : xa \in J\}$, then $b\in I$. Our purpose is to extend the concept of $r$-ideals to $H_J$-ideals of commutative rings. Then, we investigate the basic properties of $H_J$-ideals and also, we give some examples with regard to $H_J$-ideals.