On graded hyperrings and graded hypermodules

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.