On graded hyperrings and graded hypermodules

Document Type : Research Paper


Department of Mathematics, Payame Noor University (PNU), P.O.BOX 19395-3697 Tehran, Iran,


Let $G$ be a monoid with identity $e$. In this paper, first we introduce the notions of $G$-graded hyperrings, graded hyperideals and graded hyperfields in the sense of Krasner hyperring $R$. Also, we define the notion of a greded $R$-hypermodules and some examples are presented. Then we investigate graded maximal, graded prime and graded primary hyperideals of a graded hyperring $R$. Finally, we study graded maximal, graded prime and graded primary subhypermodules of a graded $R$-hypermodule $M$ and some interesting results on these concepts are given.


[1] R. Ameri and M. Norouzi, On commutative hyperring, Int. of Alg. Hyper. Appl., Vol. 1 No. 1 (2014), pp. 45-58.
[2] R. Ameri and M. Norouzi, Prime and primary hyperideals in Krasner (m; n)-hyperrings, European J. Combin., Vol. 34 (2013), pp. 379-390.
[3] R. Ameri, M. Norouzi and V. Leoreanu, On prime and primary subhypermodules of (m; n)-hypermodules, European J. Combin., Vol. 44 (2015), pp. 175-190.
[4] M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc, Vol. 282 No. 1 (1984), pp. 237-258.
[5] P. Corsini, Prolegomena of Hypergroup Theory, (second edition), Aviani Editore, (1993).
[6] P. Corsini and V. Leoreanu, Applications of Hyperstructures Theory, Adv. Math., Kluwer Academic Publishers, (2013).
[7] B. Davvaz and V. Leoreanu-Fotea, Hyperring theory and applications, Internationl Academic Press, USA, (2007).
[8] B. Davvaz and T. Vougiouklis, Commutative rings obtained from hyperrings (Hv-rings) with relations, Comm. Alg., Vol. 35 No. 11 (2007), pp. 3307-3320.
[9] F. Farzalipour and P. Ghiasvand, On the Union of Graded Prime Submodules, Thai. J. Math, Vol. 9 No. 1 (2011), pp. 49-55.
[10] F. Farzalipour and P. Ghiasvand, On graded semiprime and graded weakly semiprime ideals, Int. Elect. J. Alg., Vol. 13 (2013), pp. 15-22.
[11] M. Krasner, A class of hyperrings and hyper eld, Intern. J. Math. Math. Sci., Vol. 6 No. 2 (1983), pp. 307-312.
[12] A. Macdonal, Commutative Algebra, Addison Wesley Publishing Company, (1969).
[13] F. Marty, Sur une generalization de la notion de groupe, in: 8iem Congres Math. Scandinaves, Stockholm, (1934), pp. 45-49.
[14] N. Nastasescu and F. Van Oystaeyen, Graded Rings Theory, Mathematical Library 28, North Holand, Amsterdam, (1937).
[15] R. Rota, Strongly distributive multiplicative hyperrings, J. Geo., Vol. 39 (1990), pp. 130-138.
[16] J. P. Serre, Local Algebra, Springer Verlag, (2000).
[17] T. Vougiouklis, The fundamental relation in hyperrings. The general hyper eld, Alg. Hyper. Appl. (Xanthi,1990), Word Sci. Publishing, Teaneck, NJ, (1991), pp. 203-211.
[18] M. M. Zahedi and R. Amery, On the prime, primary and maximal subhypermodules, Italian J. Pure Appl.Math., Vol. 5 (1999), pp. 61-80.