On $1$-absorbing $δ$-primary ideals

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

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