In this paper, we study modules and filters on semihoops. Firstly, we introduce the definition of modules on semihoops and give some examples to illustrate it. Also, we get some significant results related to modules on semihoops. If the semihoop $G$ can generate an Abelian group, then $G$ is a module of any subalgebra $S$ of the semihoop $G$. Then, we use modules and filters to investigate the relationship between modules and semihoops regarding quotient algebras. Secondly, by introducing the definitions of prime submodules and torsion free modules on semihoops, we explore the relationship among prime modules, filters, and torsion free modules. Moreover, we discuss the relationship between the images and inverse images under the homomorphism of semihoops and modules, respectively. Finally, we define multiplication modules and comultiplication modules on semihoops. We study the relationship among multiplication modules and submodules on semihoops and provide the condition for comultiplication modules to satisfy the descending chain condition.

We state several conditions under which comultiplica-tion and weak comultiplication modules are cyclic and study strong comultiplication modules and comultiplication rings. In particu-lar, we will show that every faithful weak comultiplication module having a maximal submodule over a reduced ring with a  nite in-decomposable decomposition is cyclic. Also we show that if M is an strong comultiplication R-module, then R is semilocal and M is  nitely cogenerated. Furthermore, we de ne an R-module M to be p-comultiplication, if every nontrivial submodule of M is the annihilator of some prime ideal of R containing the annihila-tor of M and give a characterization of all cyclic p-comultiplication modules. Moreover, we prove that every p-comultiplication module which is not cyclic, has no maximal submodule and its annihilator is not prime. Also we give an example of a module over a Dedekind domain which is not weak comultiplication, but all of whose local-izations at prime ideals are comultiplication and hence serves as a counterexample to [11, Proposition 2. 3] and [12, Proposition 2. 4].

LetG be a group. Let R be a G -graded commutative ring with identity, and let M be a G -graded module over R. Two graded submodulesN and K of graded module M are called graded coprime whenever N+K=M. In this paper, some properties of graded coprime submodules are discussed. For example, we show that ifM is a graded finitely generated module, then two graded submodulesN and K of M are graded coprime if and only if gradM (N) and gradM (K)are graded coprime.

‎In this paper‎, ‎our aim is to introduce and study the essential submodules of an $R$-module $M$ relative to an arbitrary submodule $T$ of $M$‎. ‎Let $T$ be an arbitrary submodule of an $R$-module $M$‎, ‎then we say that a submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎whenever for every submodule $X$ of $M$‎, ‎$N\cap X\subseteq T$ implies that‎ ‎$(T:M)\subseteq ^{e}{\rm Ann}(X)$‎. ‎We investigate some new results concerning to this class of submodules‎. ‎Among various results we prove that for a faithful multiplication $R$-module $M$‎, ‎if the submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎then $(N:M)$ is an essential ideal of $R$ relative to $(T:M)$‎. ‎The converse is true if $M$ is moreover a finitely generated module‎.