Let $ R $ be a commutative ring with non-zero identity. We define a proper submodule $ N $ of an $ R $-module $ M $ to be weakly prime if $ 0\not = rm\in N $( $ r\in R, m\in M $) implies $ m\in N $ or $ rM\subseteq N $. A number of results concerning weakly prime submodules are given. For example, we give three other characterizations of weakly prime submodules.

Let G be a monoid with identity e, and let R be a G-graded commutative ring. Here we study the graded prime submodules of a graded R-module. While the bulk of this work is devoted to extending some results from prime submodules to graded prime submodules. A number of results concerning of these class of submodules are given. Mathematics Subject Classification: 13A02, 16W50

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.

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.