$n-$absorbing $I-$prime hyperideals in multiplicative hyperrings

Abstract

In this paper, we define the concept $I-$prime hyperideal in a multiplicative hyperring $R$. A proper hyperideal $P$ of $R$ is an $I-$prime hyperideal if for $a, b \in R$ with $ab \subseteq P-IP$ implies $a \in P$ or $b \in P$. We provide some characterizations of $I-$prime hyperideals. Also we conceptualize and study the notions $2-$absorbing $I-$prime and $n-$absorbing $I-$prime hyperideals into multiplicative hyperrings as generalizations of prime ideals. A proper hyperideal $P$ of a hyperring $R$ is an $n-$absorbing $I-$prime hyperideal if for $x_1, \cdots,x_{n+1} \in R$ such that $x_1 \cdots x_{n+1} \subseteq P-IP$, then $x_1 \cdots x_{i-1} x_{i+1} \cdots x_{n+1} \subseteq P$ for some $i \in \{1, \cdots ,n+1\}$. We study some properties of such generalizations. We prove that if $P$ is an $I-$prime hyperideal of a hyperring $R$, then each of $\frac{P}{J}$, $S^{-1} P$, $f(P)$, $f^{-1}(P)$, $\sqrt{P}$ and $P[x]$ are $I-$prime hyperideals under suitable conditions and suitable hyperideal $I$, where $J$ is a hyperideal contains in $P$. Also, we characterize $I-$prime hyperideals in the decomposite hyperrings. Moreover, we show that the hyperring with finite number of maximal hyperideals in which every proper hyperideal is $n-$absorbing $I-$prime is a finite product of hyperfields.