On $1$-absorbing $\delta$-primary ideals

Abstract

Let $R$ be a commutative ring with nonzero identity. Let $\mathcal{I}(R)$ be the set of all ideals of $R$ and let $\delta : \mathcal{I}(R)\longrightarrow \mathcal{I}(R)$ be a function. Then $\delta$ is called an expansion function of ideals of $R$ if whenever $L, I, J$ are ideals of R with $J \subseteq I$, we have $L \subseteq \delta( L)$ and $\delta(J)\subseteq \delta(I)$. Let $\delta$ be an expansion function of ideals of $R$. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of $\delta$-primary ideals. A proper ideal $I$ of $R$ is said to be a $1$-absorbing $\delta$-primary ideal if whenever nonunit elements $a,b,c \in R $ and $abc\in I$, then $ab \in I$ or $c\in \delta(I).$ Moreover, we give some basic properties of this class of ideals and we study the $1$-absorbing $\delta$-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.