• Title/Summary/Keyword: graded modules

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GROUP GRADED TYPES OF BÉZOUT MODULES

  • Ahmed, Mamoon;Moh'D, Fida
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.523-534
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    • 2017
  • In this paper, we introduce two group graded types of $B{\acute{e}}zout$ modules, namely graded-$B{\acute{e}}zout$ modules and weakly graded-$B{\acute{e}}zout$ modules, which are two $B{\acute{e}}zout$ versions in Graded Module Theory. We investigate the relationship among the three types of $B{\acute{e}}zout$ modules, the ordinary $B{\acute{e}}zout$ modules and the two graded types of $B{\acute{e}}zout$ modules. Also, we study the structure of these new $B{\acute{e}}zout$ modules along with different properties; for instance, "A graded-$B{\acute{e}}zout$ R-module, with R being a Noetherian ring, is Noetherien iff it is gr-Noetherian".

ON DUAL ZARISKI TOPOLOGY OVER GRADED COMULTIPLICATION MODULES

  • Abu-Dawwas, Rashid;Alshehry, Azzh Saad
    • Communications of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.11-18
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    • 2021
  • In this article, we deal with Zariski topology on graded comultiplication modules. The purpose of this article is obtaining some connections between algebraic properties of graded comultiplication modules and topological properties of dual Zariski topology on graded comultiplication modules.

GRADED w-NOETHERIAN MODULES OVER GRADED RINGS

  • Wu, Xiaoying
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1319-1334
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    • 2020
  • In this paper, we study the basic theory of the category of graded w-Noetherian modules over a graded ring R. Some elementary concepts, such as w-envelope of graded modules, graded w-Noetherian rings and so on, are introduced. It is shown that: (1) A graded domain R is graded w-Noetherian if and only if Rg𝔪 is a graded Noetherian ring for any gr-maximal w-ideal m of R, and there are only finite numbers of gr-maximal w-ideals including a for any nonzero homogeneous element a. (2) Let R be a strongly graded ring. Then R is a graded w-Noetherian ring if and only if Re is a w-Noetherian ring. (3) Let R be a graded w-Noetherian domain and let a ∈ R be a homogeneous element. Suppose 𝖕 is a minimal graded prime ideal of (a). Then the graded height of the graded prime ideal 𝖕 is at most 1.

On Graded Quasi-Prime Submodules

  • AL-ZOUBI, KHALDOUN;ABU-DAWWAS, RASHID
    • Kyungpook Mathematical Journal
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    • v.55 no.2
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    • pp.259-266
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    • 2015
  • 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 introduce the concept of graded quasi-prime submodules and give some basic results about graded quasi-prime submodules of graded modules. Special attention has been paid, when graded modules are graded multiplication, to find extra properties of these submodules. Furthermore, a topology related to graded quasi-prime submodules is introduced.

DECOMPOSITIONS OF GRADED MAXIMAL SUBMODULES

  • Moh'd, Fida
    • Communications of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.1-15
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    • 2022
  • In this paper, we present different decompositions of graded maximal submodules of a graded module. From these decompositions, we derive decompositions of the graded Jacobson radical of a graded module. Using these decompositions, we prove new theorems about graded maximal submodules, improve old theorems, and give other proofs for old theorems.

FINITELY GENERATED gr-MULTIPLICATION MODULES

  • Park, Seungkook
    • Journal of the Chungcheong Mathematical Society
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    • v.25 no.4
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    • pp.717-723
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    • 2012
  • In this paper, we investigate when gr-multiplication modules are finitely generated and show that if M is a finitely generated gr-multiplication R-module then there is a lattice isomorphism between the lattice of all graded ideals I of R containing ann(M) and the lattice of all graded submodules of M.

Notes on correct modules

  • Lee, Dong-Soo;Park, Chul-Hwan
    • Communications of the Korean Mathematical Society
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    • v.11 no.2
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    • pp.295-301
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    • 1996
  • In this paper we will define correct module and strongly correct module. We can have some basic results about those modules. And we will show that M is a graded correct R-module if and only if $M_e$ is a correct $R_e$-module.

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EQUIVARIANT CROSSED MODULES AND COHOMOLOGY OF GROUPS WITH OPERATORS

  • CUC, PHAM THI;QUANG, NGUYEN TIEN
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.4
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    • pp.1077-1095
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    • 2015
  • In this paper we study equivariant crossed modules in its link with strict graded categorical groups. The resulting Schreier theory for equivariant group extensions of the type of an equivariant crossed module generalizes both the theory of group extensions of the type of a crossed module and the one of equivariant group extensions.

GRADED PRIMAL SUBMODULES OF GRADED MODULES

  • Darani, Ahmad Yousefian
    • Journal of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.927-938
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    • 2011
  • Let G be an abelian monoid with identity e. Let R be a G-graded commutative ring, and M a graded R-module. In this paper we first introduce the concept of graded primal submodules of M an give some basic results concerning this class of submodules. Then we characterize the graded primal ideals of the idealization R(+)M.

Results of Graded Local Cohomology Modules with respect to a Pair of Ideals

  • Dehghani-Zadeh, Fatemeh
    • Kyungpook Mathematical Journal
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    • v.58 no.1
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    • pp.9-17
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    • 2018
  • Let $R ={\oplus}_{n{\in}Z}R_n$ be a graded commutative Noetherian ring and let I be a graded ideal of R and J be an arbitrary ideal. It is shown that the i-th generalized local cohomology module of graded module M with respect to the (I, J), is graded. Also, the asymptotic behaviour of the homogeneous components of $H^i_{I,J}(M)$ is investigated for some i's with a specified property.