we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

By considering the notion of multiplication modules over a commutative ring with identity, first we introduce the notion product of two submodules of such modules. Then we use this notion to characterize the prime submodules of a multiplication module. Finally, we state and prove a version of Nakayama lemma for multiplication modules and find some related basic results. 1. Introduction. Let R b...

Let R be a commutative ring with identity and M an R–module. If M is either locally cyclic projective or faithful multiplication then M is locally either zero or isomorphic to R. We investigate locally cyclic projective modules and the properties they have in common with faithful multiplication modules. Our main tool is the trace ideal. We see that the module structure of a locally cyclic proje...

Let l be a prime, and let Γ be a finite subgroup of GLn(Fl) = GL(V ). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad V there exists an element g ∈ Γ with an eigenvalue α such that tr eg,αW 6= 0. Here, eg,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the definition of adequacy in section 2. Let Γ denote...

Let l be a prime, and let Γ be a finite subgroup of GLn(Fl) = GL(V ). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad V there exists an element g ∈ Γ with an eigenvalue α such that tr eg,αW 6= 0. Here, eg,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the definition of adequacy in section 2. Let Γ denote...

Let G be an abelian group and let R be a commutative G-graded super-ring (briefly, graded super-ring) with unity 1 6= 0. We say that a ∈ h(R), where h(R) is the set of homogeneous elements in R, is weakly prime to a graded superideal I of R if 0 6= r a ∈ I , where r ∈ h(R), then r ∈ I . If ν(I ) is the set of homogeneous elements in R that are not weakly prime to I , then we define I to be weak...

In this paper, we introduce and study graded weakly 1-absorbing prime ideals in commutative rings. Let \(G\) be a group \(R\) \(G\)-graded ring with nonzero identity \(1\neq0\). A proper ideal \(P\) of is called if for each nonunits \(x,y,z\in h(R)\) \(0\neq xyz\in P\), then either \(xy\in P\) or \(z\in P\). We give many properties characterizations ideals. Moreover, investigate under homomorph...