$\delta$-$n$-ideals of commutative rings

Abstract

Let $R$ be a commutative ring with nonzero identity, and $\delta :\mathcal{I}(\mathcal{R})\to \mathcal{I}(\mathcal{R})$ be an ideal expansion where $\mathcal{I}(\mathcal{R})$ the set of all ideals of $R$. In this paper, we introduce the concept of $\delta$-$n$-ideals which is an extension of $n$-ideals in commutative rings. We call a proper ideal $I$ of $R$ a $\delta$-$n$-ideal if whenever $a,b\in R$ with$ab\in I$ and $a\notin \sqrt{0}$, then $b\in \delta (I)$. For example, ${\delta }_1$ is defined by ${\delta }_1(I)=\sqrt{I}.$ A number of results and characterizations related to $\delta$-$n$-ideals are given. Furthermore, we present some results related to quasi $n$-ideals which is for the particular case $\delta ={\delta }_1.$