Almost graded multiplication and almost graded comultiplication modules

Malik Bataineh; Rashid Abu-Dawwas; Jenan Shtayat

doi:10.1515/dema-2020-0023

Almost graded multiplication and almost graded comultiplication modules

Demonstratio Mathematica ◽

10.1515/dema-2020-0023 ◽

2020 ◽

Vol 53 (1) ◽

pp. 325-331

Author(s):

Malik Bataineh ◽

Rashid Abu-Dawwas ◽

Jenan Shtayat

Keyword(s):

Commutative Ring ◽

Graded 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.

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ON GRADED S-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.10.11.8 ◽

2021 ◽

Vol 10 (11) ◽

pp. 3479-3489

Author(s):

K. Al-Zoubi ◽

M. Al-Azaizeh

Keyword(s):

Abelian Group ◽

Commutative Ring ◽

Closed Subset ◽

Commutative Rings ◽

Graded Modules ◽

Prime Submodules ◽

Prime Submodule

Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity, $M$ a graded $R$-module and $S\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $S$-prime submodules of graded modules over graded commutative rings. We investigate some properties of this class of graded submodules and their homogeneous components. Let $N$ be a graded submodule of $M$ such that $(N:_{R}M)\cap S=\emptyset $. We say that $N$ is \textit{a graded }$S$\textit{-prime submodule of }$M$ if there exists $s_{g}\in S$ and whenever $a_{h}m_{i}\in N,$ then either $s_{g}a_{h}\in (N:_{R}M)$ or $s_{g}m_{i}\in N$ for each $a_{h}\in h(R) $ and $m_{i}\in h(M).$

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On graded Ie-prime submodules of graded modules over graded commutative rings

Publications de l Institut Mathematique ◽

10.2298/pim2124047a ◽

2021 ◽

Vol 110 (124) ◽

pp. 47-55

Author(s):

Shatha Alghueiri ◽

Khaldoun Al-Zoubi

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Graded Modules ◽

Prime Submodules ◽

Prime Submodule

Let G be a group with identity e. Let R be a G-graded commutative ring with identity and M a graded R-module. We introduce the concept of graded Ie-prime submodule as a generalization of a graded prime submodule for I =?g?G Ig a fixed graded ideal of R. We give a number of results concerning this class of graded submodules and their homogeneous components. A proper graded submodule N of M is said to be a graded Ie-prime submodule of M if whenever rg ? h(R) and mh ? h(M) with rgmh ? N ? IeN, then either rg ? (N :R M) or mh ? N.

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

Proyecciones (Antofagasta) ◽

10.22199/issn.0717-6279-3467 ◽

2021 ◽

Author(s):

Khaldoun Falah Al-Zoubi ◽

Mohammed Al-Dolat

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Graded Modules ◽

Primary Ideals

Let G be a group with identity e. Let R be a G-graded commutative ring andM a graded R-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 and extra properties of these graded submodules.

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On graded P-compactly packed modules

Open Mathematics ◽

10.1515/math-2015-0045 ◽

2015 ◽

Vol 13 (1) ◽

Author(s):

Khaldoun Al-Zoubi ◽

Imad Jaradat ◽

Mohammed Al-Dolat

Keyword(s):

Commutative Ring ◽

Graded Modules

AbstractLet 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 P-compactly packed modules and we give a number of results concerning such graded modules. In fact, our objective is to investigate graded P-compactly packed modules and examine in particular when graded R-modules are P-compactly packed. Finally, we introduce the concept of graded finitely P-compactly packed modules and give a number of its properties.

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Some properties of graded 2-prime submodules

Asian-European Journal of Mathematics ◽

10.1142/s1793557115500163 ◽

2015 ◽

Vol 08 (02) ◽

pp. 1550016

Author(s):

Khaldoun Al-Zoubi

Keyword(s):

Commutative Ring ◽

Graded Modules ◽

Prime Submodules ◽

Prime Submodule

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 2-prime submodule and we give a number of results concerning such modules. Also, we introduce and prove the graded 2-prime avoidance theorem for graded modules.

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

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.41917 ◽

2021 ◽

Vol 39 (4) ◽

pp. 57-64 ◽

Cited By ~ 1

Author(s):

Khaldoun falah Al-Zoubi ◽

Rweili Alkhalaf

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Graded Modules ◽

Primary Submodules

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-primary submodules of graded modules over graded commutative rings. Various properties of graded quasi-primary submodules are considered.

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On quasi-radical of near-ring of polynomials

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2019.56.2.1430 ◽

2019 ◽

Vol 56 (2) ◽

pp. 252-259

Author(s):

Ebrahim Hashemi ◽

Fatemeh Shokuhifar ◽

Abdollah Alhevaz

Keyword(s):

Commutative Ring ◽

Nilpotent Elements ◽

Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

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N-ideals of commutative rings

Filomat ◽

10.2298/fil1710933t ◽

2017 ◽

Vol 31 (10) ◽

pp. 2933-2941 ◽

Cited By ~ 3

Author(s):

Unsal Tekir ◽

Suat Koc ◽

Kursat Oral

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Proper Ideal ◽

Prime Ideals ◽

Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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