### On φ-S-1-absorbing δ-primary ideals of commutative rings

#### Abstract

Let $R$ be a commutative ring with unity $(1\not=0)$ and let $\mathfrak{J}(R)$ be the set of all ideals of $R$. Let $\phi:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)\cup\{\emptyset\}$ be a reduction function of ideals of $R$ and let $\delta:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)$ be an expansion function of ideals of $R$. We recall that a proper ideal $I$ of $R$ is called a $\phi$-1-absorbing $\delta$-primary ideal of $R$, if whenever $abc\in I-\phi(I)$ for some nonunit elements $a, b, c\in R$, then $ab\in I$ or $c\in\delta(I)$. In this paper, we introduce a new class of ideals that is a generalization to the class of $\phi$-1-absorbing $\delta$-primary ideals. Let $S$ be a multiplicative subset of $R$ such that $1\in S$ and let $I$ be a proper ideal of $R$ with $S\cap I=\emptyset$, then $I$ is called a $\phi$-$S$-1-absorbing $\delta$-primary ideal of $R$ associated to $s\in S,$ if whenever $abc\in I-\phi(I)$ for some nonunit elements $a, b, c\in R$, then $sab\in I$ or $sc\in\delta(I)$. In this paper, we have presented a range of different examples, properties, characterizations of this new class of ideals.

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