(Weakly) $(\alpha,\beta)$-prime hyperideals in commutative multiplicative hypeering

TL;DR

This paper introduces and studies (weakly) $(eta,eta)$-prime hyperideals in commutative multiplicative hyperrings, exploring their properties and extending the concept of prime hyperideals in hyperring theory.

Contribution

It defines (weakly) $(eta,eta)$-prime hyperideals and investigates their properties within commutative multiplicative hyperrings, extending existing hyperring theory.

Findings

Characterization of (weakly) $(eta,eta)$-prime hyperideals

Properties and conditions for hyperideals to be (weakly) $(eta,eta)$-prime

Extension of prime hyperideal concepts to hyperring context

Abstract

Let $H$ be a commutative multiplicative hyperring and $\alpha, \beta \in \mathbb{Z}^+$. A proper hyperideal $P$ of $H$ is called (weakly) $(\alpha,\beta)$-prime if $x^\alpha \circ y \subseteq P$ for $x,y \in H$ implies $x^\beta \subseteq P$ or $y \in P$. In this paper, we aim to investigate (weakly) $(\alpha,\beta)$-prime hyperideals and then we present some properties of them.