$\varphi$-$\delta$-Primary Hyperideals in Krasner Hyperrings

Abstract

In this paper, we study commutative Krasner hyperring with nonzero identity. $\varphi$-prime, $\varphi$-primary and $\varphi$-$\delta$-primary hyperideals are introduced. We intend to extend the concept of $\delta$-primary hyperideals to $\varphi$-$\delta$-primary hyperideals. We give some characterizations of hyperideals to classify them. We denote the set of all hyperideals of $\Re$ by $L(\Re )$ (all proper hyperideals of $\Re$ by $L^{\ast }(\Re )).$ Let $\varphi$ be a reduction function such that $\varphi :L(\Re )\to L(\Re )\cup \{\varnothing \}$ and $\delta$ be an expansion function such that $\delta :L(\Re )\to L(\Re ).$ $N$ be a proper hyperideal of $\Re .$ $N$ is called $\varphi$-$\delta$-primary hyperideal of $\Re$ if $a\circ b\in N-$ $\varphi (N),$ then $a\in N$ or $b\in \delta (N),$ for some $a,b\in \Re .$ We\ discuss the relation between $\varphi$-$\delta$-primary hyperideal and other hyperideals.