Document Type : Original Manuscript


Mathematics Department, Faculty of Science, Soran University, P.O. Box 44008, Soran, Erbil Kurdistan Region, Iraq.


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.


 1. I. Akray, I-prime ideals, J. Algebra Relat. Topics, 4(2) (2016), 41–47.
2. I. Akray and M. B. Mrakhan, n-Absorbing I-ideals, Khayyam J. Math., 6(2) (2020), 174–179.
3. M. Anbarloei, 2-Prime Hyperideals of Multiplicative Hyperrings, J. Math., (2022), 1–9.
4. A. Badawi, On 2-absorbing of commutative rings, Bull. Aust. Math. Soc., 75(3) (2007), 417–429.
5. A. Badawi and D. F. Anderson, On n-absorbing ideals of commutative rings, Comm. Algebra, 39(5) (2011), 1646–1672.
6. R. P. Ciampi and R. Rota, Polynomials over multiplicative hyperrings, J. Discrete Math. Sci. Cryptogr., 6(2-3) (2013), 217–225.
7. U. Dasgupta, On prime and primary hyperideals of a multiplicative hyperring, Annals of the Alexandru Ioan Cuza University-Mathematics, 58(1) (2012), 19–36.
8. M. De Salvo and G.L. Faro, On the n*-complete hypergroups, Discrete Math., 208 (1999), 177–188.
9. J. Huckaba, Commutative rings with zero divisors, Marcel Dekker, New York, Basel, 1988.
10. M. Krasner, A class of hyperrings and hyperfields, Int. J. Math. Math. Sci., 6(2) (1983), 307–311.
11. F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandenaves, Stockholm, (1934), 45–49.
12. R. Procesi and R. Rota, On some classes of hyperstructures, Discrete Math., 208 (1999), 485–497.
13. R. Rota, Sugli iperanelli moltiplicativi, Rend. Di Math., Series VII, 2(4) (1982), 711–724.
14. G. Ulucak, On expansions of prime and 2-absorbing hyperideals in multiplicative hyperrings, Turkish J. Math., 43(3) (2019), 1504–1517.
15. T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, 1994.
16. H. S. Wall, Hypergroups, Amer. J. Math., 59(1) (1937), 77–98