Let [Formula: see text] be an abelian group with identity [Formula: see text]. Let [Formula: see text] be a graded multiplicative hyperring and [Formula: see text] be a function where [Formula: see text] is the set of graded hyperideals of [Formula: see text] and [Formula: see text] for every graded hyperideal [Formula: see text] of [Formula: see text]. In this paper, we introduce and study the concepts of graded [Formula: see text]-[Formula: see text]-absorbing hyperideals and graded [Formula: see text]-[Formula: see text]-absorbing primary hyperideals of [Formula: see text]. Moreover, we give a number of main results and the basic properties concerning these classes of graded hyperideals and their homogeneous components.

Abstract. Let G be a group with identity e, and let R be a G-graded commutative ring, and let M be a graded R-module. In this paper we characterize graded weak multiplication modules.

Let G be an abelian group with identity e. Let R be a graded multiplicative hyperring and ? : Igr(R) ? Igr(R) be an expansion function of Igr(R), where Igr(R) is the set of all graded hyperideals of R. In this paper, we introduce and study the concepts of graded ?-primary hyperideals of R and graded 2-absorbing ?-primary hyperideals of R which are the extended classes of graded prime and graded 2- absorbing hyperideals of R, respectively. Moreover, we give the basic properties of these new types of graded hyperideals and investigate the relations among these structures.

The authors introduce the concept of almost semiprime subsemimodules of semimodules over a commutative semiring R. They investigated some basic properties of almost semiprime and weakly semiprime subsemimodules and gave some characterizations of them, especially, for (fnitely generated faithful) multiplication semimodules. They also study the relations among the semiprime, weakly semiprime and almost semiprime subsemimodules of semimodules over semirings.

Let G be a group with identity e. Let R be a commutative Ggraded ring with non-zero identity, S ⊆ h(R) a multiplicatively closed subset of R and M a graded R-module. In this article, we introduce and study the concept of graded S-1-absorbing prime submodules. A graded submodule N of M with (N : R M ) ∩ S = ∅ is said to be graded S-1-absorbing prime, if there exists an sg ∈ S such that whenever a h b h m k ∈ N , then either. Some examples, characterizations and properties of graded S-1-absorbing prime submodules are given. Moreover, we give some characterizations of graded S-1-absorbing prime submodules in graded multiplicative modules.