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

TL;DR

This paper introduces and studies a new class of ideals called $\,\phi$-$\delta$-$S$-primary ideals in commutative rings, generalizing existing $\,\phi$-$\delta$-primary ideals, with various properties and examples.

Contribution

The paper defines $\,\phi$-$\delta$-$S$-primary ideals, extending the concept of $\,\phi$-$\delta$-primary ideals, and explores their properties and characterizations.

Findings

Introduced the concept of $\,\phi$-$\delta$-$S$-primary ideals.

Provided examples illustrating the new class.

Established properties and characterizations of these ideals.

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 $\phi$-$\delta$-primary ideal of $R$ if whenever $a,b\in R$ and $ab\in I-\phi(I)$, 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 $\phi$-$\delta$-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 $\phi$-$\delta$-$S$-primary ideal of $R$ associated to $s\in S$ if whenever $a,b\in R$ and $ab\in I-\phi(I)$, 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.