$\delta$-$r$-Hyperideals and $\phi$-$\delta$-$r$-Hyperideals of Commutative Krasner Hyperrings

TL;DR

This paper introduces and explores the concepts of $ $-hyperideals and $$-$ $-hyperideals in commutative Krasner hyperrings, extending the theory of hyperideals with new definitions and properties.

Contribution

It defines $ $-hyperideals and $$-$ $-hyperideals in Krasner hyperrings, extending existing hyperideal concepts with new properties and examples.

Findings

Defined $ $-hyperideals and $$-$ $-hyperideals in Krasner hyperrings.

Established properties and provided examples of these hyperideals.

Extended the theory of hyperideals in algebraic structures.

Abstract

In this paper, our purpose is to define the expansion of $r$-hyperideals and extend this concept to $\phi$-$\delta$-$r$-hyperideal. Let $\Re$ be a commutative Krasner hyperring with nonzero identity. Given an expansion $\delta$ of hyperideals, a proper hyperideal $N$ of $\Re$ is called $\delta $-$r$-hyperideal if $a\cdot b\in N$ with $ann(a)=0$ implies that $b\in \delta(N)$, for all $a,b\in\Re$. Therefore, given an expansion $\delta$ of hyperideals and a hyperideal reduction $\phi$, a proper hyperideal $N$ of $\Re$ is called $\phi$-$\delta$-$r$-hyperideal if $a\cdot b\in N-\phi(N)$ with $ann(a)=0$ implies that $b\in\delta(N)$, for all $a,b\in\Re$. We investigate some of their properties and give some examples.