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

Abstract

Let $R$ be a commutative ring with nonzero identity, and $\delta :\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ 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 $\delta=\delta_{1}.$