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

Abstract

In this paper, we study commutative Krasner hyperring with nonzero identity. $\phi$-prime, $\phi$-primary and $\phi$-$\delta$-primary hyperideals are introduced. We intend to extend the concept of $\delta$-primary hyperideals to $\phi$-$\delta$-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 $\phi$ be a reduction function such that $\phi:L(\Re)\rightarrow L(\Re)\cup\{\emptyset\}$ and $\delta$ be an expansion function such that $\delta:L(\Re)\rightarrow L(\Re).$ $N$ be a proper hyperideal of $\Re.$ $N$ is called $\phi$-$\delta$-primary hyperideal of $\Re$ if $a\circ b\in N-$ $\phi(N),$ then $a\in N$ or $b\in\delta(N),$ for some $a,b\in\Re.$ We\ discuss the relation between $\phi$-$\delta$-primary hyperideal and other hyperideals.