On $\varphi$-$\delta$-S-primary ideals of commutative rings

Abstract

Let $R$ be a commutative ring with unity $(1\ne 0)$ and let $\mathfrak{J}(R)$ be the set of all ideals of $R$. Let $\varphi :\mathfrak{J}(R)\to \mathfrak{J}(R)\cup \{\varnothing \}$ be a reduction function of ideals of $R$ and let $\delta :\mathfrak{J}(R)\to \mathfrak{J}(R)$ be an expansion function of ideals of $R$. We recall that a proper ideal $I$ of $R$ is called a $\varphi$-$\delta$-primary ideal of $R$ if whenever $a,b\in R$ and $ab\in I-\varphi (I)$, then $a\in I$ or $b\in \delta (I)$. In this paper, we introduce a new class of ideals that is a generalization to the class of $\varphi$-$\delta$-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 $S\cap I=\varnothing$, then $I$ is called a $\varphi$-$\delta$-$S$-primary ideal of $R$ associated to $s\in S$ if whenever $a,b\in R$ and $ab\in I-\varphi (I)$, then $sa\in I$ or $sb\in \delta (I)$. In this paper, we have presented a range of different examples, properties, characterizations of this new class of ideals.