n−ABSORBING I−PRIME HYPERIDEALS IN MULTIPLICATIVE HYPERRINGS

In this paper, we define the concept I−prime hyper ideal in a multiplicative hyperring R. A proper hyperideal P of R is an I−prime hyperideal if for a, b ∈ R with ab ⊆ P − IP implies a ∈ P or b ∈ P. We provide some characterizations of I−prime hyperideals. Also we conceptualize and study the no tions 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 x1, · · · , xn+1 ∈ R such that x1 · · · xn+1 ⊆ P −IP, then x1 · · · xi−1xi+1 · · · xn+1 ⊆ P for some i ∈ {1, · · · , 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 P/J , S^−1P, f(P), f^−1 (P), √ P and P[x] are I−prime hyperideals under suit able 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 fields.