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

TL;DR

This paper introduces and studies the concept of $\,\phi$-$\delta$-primary hyperideals in commutative Krasner hyperrings, extending existing notions and providing characterizations and relations among hyperideals.

Contribution

It extends the concept of $\,\delta$-primary hyperideals to $\,\phi$-$\delta$-primary hyperideals and characterizes their properties in Krasner hyperrings.

Findings

Defined $\,\phi$-$\delta$-primary hyperideals in Krasner hyperrings.

Provided characterizations and relations with other hyperideals.

Extended the theory of hyperideals in hyperring algebra.

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.