$δ$-$r$-Hyperideals and $ϕ$-$δ$-$r$-Hyperideals of Commutative Krasner Hyperrings

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

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