Comultiplication Modules Research Articles

On graded S−comultiplication modules

In this paper, we introduce the concept of graded S−comultiplication modules. Several results concerning graded S−comultiplication modules are proved. We show that N is a graded S−second submodule of a graded S−comultiplication R−module M if and only if Ann_R(N) is a graded S−prime ideal of R and there exists x ∈ S such that xN ⊆x- for every x- ∈ S.

Finitely generated coreduced comultiplication modules

This paper deals with some results concerning finitely generated coreduced comultiplication modules over a commutative ring.

Classification of 1-absorbing comultiplication modules over a pullback ring

One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring R. In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.

On $S$-comultiplication modules

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is a multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion of an $S$-prime submodule which is a generalization of a prime submodule and used them to characterize certain classes of rings/modules such as prime submodules, simple modules, torsion free modules,\ $S$-Noetherian modules and etc. Afterwards, in \cite{AnArTeKo}, Anderson et al. defined the concepts of $S$-multiplication modules and $S$-cyclic modules which are $S$-versions of multiplication and cyclic modules and extended many results on multiplication and cyclic modules to $S$-multiplication and $S$-cyclic modules. Here, in this article, we introduce and study $S$-comultiplication modules which are the dual notion of $S$-multiplication module. We also characterize certain classes of rings/modules such as comultiplication modules, $S$-second submodules, $S$-prime ideals and $S$-cyclic modules in terms of $S$-comultiplication modules. Moreover, we prove $S$-version of the dual Nakayama's Lemma.

On spectrum of comultiplication modules

In this article, comultiplication modules in terms of dual Zariski topology are characterized. Some connections between subspaces of a complement dual Zariski topology and submodules of a comultiplication module are studied. Then, some useful relations between open subsets of a complement dual Zariski topology and second radical submodules related to these are obtained. Finally, direct connections among rings, comultiplication modules, the complement dual Zariski topology and the dual Zariski topology are established.

Almost graded multiplication and almost graded comultiplication modules

AbstractLet G be a group with identity e, R be a G-graded commutative ring with a nonzero unity 1 and M be a G-graded R-module. In this article, we introduce and study the concept of almost graded multiplication modules as a generalization of graded multiplication modules; a graded R-module M is said to be almost graded multiplication if whenever a\in h(R) satisfies {\text{Ann}}_{R}(aM)={\text{Ann}}_{R}(M), then (0{:}_{M}a)=\{0\}. Also, we introduce and study the concept of almost graded comultiplication modules as a generalization of graded comultiplication modules; a graded R-module M is said to be almost graded comultiplication if whenever a\in h(R) satisfies {\text{Ann}}_{R}(aM)={\text{Ann}}_{R}(M), then aM=M. We investigate several properties of these classes of graded modules.

Pure Elements and Dual Notions of Prime Elements in Lattice Modules

This paper deals with the pure elements and the dual notions of prime elements (that is, second elements). For this, it introduces the definitions of second element and coprime element. Then it is shown that the concepts of the second element and coprime element are equivalent. Moreover, this study gives us a characterization of comultiplication modules. Finally, it defines pure elements and obtains the relation among pure, idempotent and multiplication elements.

S(0)-Weakly second submodules

We introduce the dual notions of [Formula: see text]-weakly prime submodules, that is, [Formula: see text]-weakly second submodules in a commutative ring with identity. We investigate the properties of [Formula: see text]-weakly second submodules and obtain some related useful characterizations of this dualization. Moreover, we also prove some properties of [Formula: see text]-weakly second submodules related to multiplication and comultiplication modules.

Dual Zariski Topology on Comultiplication Modules

This paper deals with dual Zariski topology on comultiplication modules. We define a subspace topology of dual Zariski topology on comultiplication modules and study some properties of this subspace topology. We prove that XsN is an Artinian topological space if and only if M satisfies the SN-condition.

Graded weak comultiplication modules

Let $G$ be a group with identity $e$, $R$ be a $G$-graded ring and $M$ be a $G$-graded $R$-module. In this article, we introduce the concept of graded weak comultiplication modules. A graded $R$-module $M$ is said to be graded weak comultiplication if for every graded prime $R$-submodule $N$ of $M$, $N=(0:_{M}I)$ for some graded ideal $I$ of $R$. We study graded weak comultiplication modules and give several results.

Comultiplication modules relative to a hereditary torsion theory

Let R be a commutative ring with identity and τ be a hereditary torsion theory on R-Mod. In this article, we introduce and study the concept of τ-comultiplication module. We present several properties and characterizations of τ-comultiplication modules. We also investigate modules for which every submodule has a unique τ-pseudo-complement and prove that every τ-comultiplication module is a module with unique τ-pseudo-complements.

ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING

The purpose of this paper is to introduce the concept of 2-absorbing comultiplication modules, which form a subclass of the class of pure-injective modules over pullback rings. A full description of all indecomposable 2- absorbing comultiplication modules with nite-dimensional top over the pull- back of two discrete valuation domains with the same residue eld is given.

Some properties of graded comultiplication modules

Abstract Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring.

Comultiplication Modules over Noncommutative Rings

Comultiplication Modules over Noncommutative Rings

Modules whose submodules are annihilators

In this paper we investigate the comultiplication modules over not necessarily commutative rings. This research was supported by the Russian Foundation for Basic Research, grant 08-01- 00693a.

On the Dual Notion of Multiplication Modules

This paper deals with some results concerning the dual notion of multiplication modules (i.e., comultiplication modules) over a commutative ring.

On multiplication and comultiplication modules

This article deals with some results concerning multiplication and comultiplication modules over a commutative ring.

On the Dual Notion of Prime Submodules (II)

Let R be a commutative ring and let M be an R-module. In this paper, we study the dual notion of prime submodules (that is, second submodules of M). Also we introduce the dual notion of weak multiplication R-modules (that is, weak comultiplication modules) in terms of second submodules and investigate some related results.

Comultiplication modules over a pullback of Dedekind domains

First, we give complete description of the comultiplication modules over a Dedekind domain. Second, if R is the pullback of two local Dedekind domains, then we classify all indecomposable comultiplication R-modules and establish a connection between the comultiplication modules and the pure-injective modules over such domains.