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

Abstract

In this paper, our purpose is to define the expansion of $r$-hyperideals and extend this concept to $\varphi$-$\delta$-$r$-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 $\varphi$, a proper hyperideal $N$ of $\Re$ is called $\varphi$-$\delta$-$r$-hyperideal if $a\cdot b\in N-\varphi (N)$ 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.