Multiplication Modules and the Ideal (original) (raw)

Graded multiplication modules and the graded ideal θ g M)

Turkish Journal of Mathematics, 2011

Let G be a group and let R be a G-graded commutative ring. For a graded R-module M , the notion of the associated graded ideal θg(M) of R is defined. It is proved that the graded ideal θg(M) is important in the study of graded multiplication modules. Among various application given, the following results are proved: if M is a graded faithful multiplication module, then θg(M) is an idempotent graded multiplication ideal of R such that θg(θg(M)) = θg(M) , and every graded representable multiplication R-module is finitely generated.

Some properties of graded comultiplication modules

Open Mathematics, 2017

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.

A Generalization of Group-Graded Modules

Symmetry

In this article, we generalize the concept of group-graded modules by introducing the concept of G-weak graded R-modules, which are R-modules graded by a set G of left coset representatives, where R is a G-weak graded ring. Moreover, we prove some properties of these modules. Finally, results related to G-weak graded fields and their vector spaces are deduced. Many considerable examples are provided with more emphasis on the symmetric group S3 and the dihedral group D6, which gives the group of symmetries of a regular hexagon.

Modules Graded byG-Sets: Duality and Finiteness Conditions

Journal of Algebra, 1997

An efficient technique to uncover the relationship between the representations of a group G and the representations of a subgroup H is to Ž . consider gradings by GrH , the class of the left H-cosets of G. This l w x method of study was initiated by Dade in 3 , where he describes the structure of the simple objects in the category R᎐gr of G-graded R-modules for a G-graded ring R, as a generalization of the classical Clifford theory. A systematic investigation of modules graded by G-sets was started w x w x by Nastasescu et al. in 9 , and then continued in other papers 1, 8, 5 .˘Ž . Categorical properties of the category G, A, R ᎐gr of A-graded left R-modules, where R is a G-graded ring and A a left G-set, were studied first, and later a series of finiteness conditions for the objects of this category were considered.

Graded Prime Submodules Over Multiplication Modules

Indian Journal of Pure and Applied Mathematics

Let G be an abelian group with identity e, R be a G-graded commutative ring and M a graded R-module where all modules are unital. Various generalizations of graded prime ideals have been studied. For example, a proper graded ideal I is a graded weakly (resp., almost) prime ideal if 0≠ab∈I (resp., ab∈I-I 2 ) implies a∈I or b∈I. Throughout this work, we define that a proper graded submodule N of M is a graded almost prime if am∈N-(N:M)N implies a∈(N:M) or m∈N. We show that graded almost prime submodules enjoy analogs of many of the properties of prime submodules.

On Graded Second and Coprimary Modules and Graded Secondary Representations

Bulletin of the Malaysian Mathematical Sciences Society, 2014

In this paper we introduce and study the concepts of graded second (gr-second) and graded coprimary (gr-coprimary) modules which are di¤erent from second and coprimary modules over arbitrary graded rings. We list some properties and characterizations of gr-second and gr-coprimary modules and also study graded prime submodules of modules with gr-coprimary decompositions. We also deal with graded secondary representations for graded injective modules over commutative graded rings. By using the concept ofsuspension ()M of a graded module M; we prove that a graded injective module over a commutative graded Noetherian ring has a graded secondary representation.

On graded primary-like submodules of graded modules over graded commutative rings

arXiv: Commutative Algebra, 2020

Let GGG be a group with identity eee. Let RRR be a GGG-graded commutative ring and MMM a graded RRR-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to find extra properties of these graded submodules.