n−ABSORBING I−PRIME HYPERIDEALS IN MULTIPLICATIVE HYPERRINGS

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.

n−ABSORBING I−PRIME HYPERIDEALS IN MULTIPLICATIVE HYPERRINGS Keywords:

multiplicative hyperring , prime hyperideal , I-prime hyperideal