(weakly) $(\alpha,\beta)$-prime hyperideals in commutative multiplicative hyperrings

Abstract

‎Let $H$ be a commutative multiplicative hyperring and $\alpha,\beta \in \mathbb{Z}^+$‎. The purpose of this paper is to introduce an intermediate class between prime hyperideals and ‎$‎(\alpha,\beta)‎$‎-closed hyperideals called ‎$‎(\alpha,\beta)‎$‎-prime hyperideals. Moreover, we study the notion of weakly ‎$‎(\alpha,\beta)‎$‎-prime hyperideals as an extension of the $(\alpha,\beta)$-prime hyperideals and a subclass of the weakly $‎(\alpha,\beta)‎$‎-closed hyperideals. We say that ‎a proper hyperideal $P$ of $H$ is (weakly) $(\alpha,\beta)$-prime if ($0 \notin x^{\alpha} \circ y \subseteq P$ ) $x^{\alpha} \circ y \subseteq P$ for $x,y \in H$ implies $x^{\beta} \subseteq P$ or ‎$‎y \in P‎$‎‎. A number of properties and results concerning them will be discussed.