n−ABSORBING I−PRIME HYPERIDEALS IN MULTIPLICATIVE HYPERRINGS

Document Type : Original Manuscript

Authors

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

Abstract

In this paper, we define the concept Iprime hyperideal in a multiplicative hyperring R. A proper hyperideal P of R is an Iprime hyperideal if for a,bR with abPIP implies aP or bP. We provide some characterizations of Iprime hyperideals. Also we conceptualize and study the notions 2absorbing Iprime and nabsorbing Iprime hyperideals into multiplicative hyperrings as generalizations of prime ideals. A proper hyperideal P of a hyperring R is an nabsorbing Iprime hyperideal if for x1,,xn+1R such that x1xn+1PIP, then x1xi1xi+1xn+1P for some i{1,,n+1}. We study some properties of such generalizations. We prove that if P is an Iprime hyperideal of a hyperring R, then each of PJ, S1P, f(P), f1(P), P and P[x] are Iprime hyperideals under suitable conditions and suitable hyperideal I, where J is a hyperideal contains in P. Also, we characterize Iprime hyperideals in the decomposite hyperrings. Moreover, we show that the hyperring with finite number of maximal hyperideals in which every proper hyperideal is nabsorbing Iprime is a finite product of hyperfields.

Keywords


 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