$r$-Hyperideals and Generalizations of $r$-Hyperideals in Krasner Hyperrings

TL;DR

This paper explores properties and generalizations of r-hyperideals in commutative Krasner hyperrings, including their relations with prime hyperideals and behavior under images and inverse images.

Contribution

It introduces a new generalization of r-hyperideals and studies their fundamental properties and relationships within Krasner hyperrings.

Findings

r-hyperideals are closed under image and inverse image operations

A new class of generalized r-hyperideals is defined and characterized

Relations between prime hyperideals and r-hyperideals are established

Abstract

In this study, we examine some properties of $r$-hyperideals in the commutative Krasner hyperrings. Some properties of $pr$-hyperideals are also studied. The relation between prime hyperideals and $r$-hyperideals is investigated. We show that the image and the inverse image of an $r$-hyperideal is also an $r$-hyperideal. We also introduce a generalization of r-hyperideals and we prove some properties of them.