arXiv subjects

Khaldoun Al-Zoubi

Publications and source records attributed to Khaldoun Al-Zoubi.

13 recordsLinked to original sources

arXiv2023

On graded weakly $J_{gr}$-semiprime submodules

Let $Γ$ be a group, $\Re$ be a $Γ$-graded commutative ring with unity $1$ and $\Im$ a graded $\Re$-module. In this paper, we introduce the concept of graded weakly $J_{gr}$-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly $J_{gr}$% -semiprime submodules. For example, we give a characterization of graded weakly $J_{gr}$-semiprime submodules. Also, we find some relations between graded weakly $J_{gr}$-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly $J_{gr}$-semiprime submodules are investigated. A proper graded submodule $U$ of $\Im$ is said to be a graded weakly $J_{gr}$-semiprime submodule of $\Im$ if whenever $r_{g}\in h(\Re),$ $m_{h}\in h(\Im)$ and $n\in %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion ^{+}$ with $0\neq r_{g}^{n}m_{h}\in U$, then $r_{g}m_{h}\in U+J_{gr}(\Im)$, where $J_{gr}(\Im)$ is the graded Jacobson radical of $\Im.$

math.GM↗

arXiv2023

On $gr$-quasi-semiprime submodules

Let $G$ be a group. A ring $R$ is called a graded ring (or $G$-graded ring) if there exist additive subgroups $R_{α}$ of $R$ indexed by the elements $α\in G$ such that $R=\bigoplus_{α\in G}R_{α}$ and $R_{α}R_{β}\subseteq R_{αβ}$ for all $α$, $% β\in G$. If an element of $R$ belongs to $h(R)=\cup _{α\in G}R_{α}$, then it is called a homogeneous. A Left $R$-module $M$ is said to be \textit{a graded }$R$\textit{-module} if there exists a family of additive subgroups $\{M_{α}\}_{α\in G}$ of $M$ such that $% M=\bigoplus_{α\in G}M_{α}$ and $R_{α}M_{β}\subseteq M_{αβ}$ for all $α,β\in G.$ Also if an element of $M$ belongs to $\cup _{α\in G}M_{α}=h(M)$, then it is called a homogeneous. A submodule $N$ of $M$ is said to be \textit{a graded submodule of }$M$ if $N=\bigoplus_{α\in G}(N\cap M_{α}):=\bigoplus_{α\in G}N_{α}$. Let $G$ be a group with identity $e$. Let $R$ be a $G$% -graded commutative ring and $M$ a graded $R$-module. A proper graded submodule $S$ of $M$ is said to be \textit{a graded semiprime (}shortly $gr$% \textit{-semiprime) submodule} if whenever $r^{n}m\in S$ where $r\in h(R)$, $% m\in h(M)$ and $n\in Z^{+}$, then $rm\in S.$ In this work, we introduce the concept of graded quasi-semiprime (shortly $gr$-quasi-semiprime) submodule as a generalization of $gr$-semiprime submodule and give some basic properties of these classes of graded submodules. We say that a proper graded submodule $S$ of $M$ is a $gr$-quasi-semiprime submodule if $% (S:_{R}M)=\{r\in R:rM\subseteq S\}$ is a $gr$-semiprime ideal of $R$.

math.AC↗

arXiv2022

On $Gr$-$C$-$2^{A}$-secondary submodules

Let $Ω$ be a group with identity $e$, $Γ$ be a $Ω$-graded commutative ring and $\Im$ a graded $Γ$-module. In this article, we introduce the concept of $gr$-$C$-$2^{A}$-secondary submodules and investigate some properties of this new class of graded submodules. A non-zero graded submodule $S$ of $\Im$ is said to be a $gr$-$C$-$2^{A}$-secondary submodule if whenever $r,s \in h(Γ)$, $L$ is a graded submodule of $\Im$, and $rs\,S\subseteq L$, then either $r\,S\subseteq L$ or $s\,S \subseteq L$ or $rs \in Gr(Ann_Γ(S))$.

math.GM↗

arXiv2022

The Zariski topology on the secondary like spectrum of a module

Let $R$ be a commutative ring with unity and $M$ be a left $R$-module. We define the secondary-like spectrum of $M$ to be the set of all secondary submodules $K$ of $M$ such that $Ann_R(soc(K))=\sqrt{Ann_R(K)}$, and we denote it by $Spec^L(M)$. In this paper, we introduce a topology on $Spec^L(M)$ having the Zariski topology on the second spectrum $Spec^s(M)$ as a subspace topology, and study several topological structures of this topology.

math.GM↗

arXiv2022

Graded modules with Noetherian graded second spectrum

Let $R$ be a $G$ graded commutative ring and $M$ be a $G$-graded $R$-module. The set of all graded second submodules of $M$ is denoted by $Spec_G^s(M)$ and it is called the graded second spectrum of $M$. In this paper, we discuss graded rings with Noetherian graded prime spectrum and obtain some conclusions. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. Using these conclusions and properties, we also investigate $Spec_G^s(M)$ with the Zariski topology from the viewpoint of being a Noetherian space and give some related outcomes.

math.AC↗

arXiv2022

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 $x\in S$ such that $xN\subseteq\bar{x}N$ for every $\bar{x}\in S$.

math.GM↗

arXiv2021

The Zariski topology on the graded primary spectrum of a graded module over a graded commutative ring

Let $R$ be a $G$-graded ring and M be a $G$-graded $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to be the set of all graded primary submodules $Q$ of M such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we define a topology on $\mathcal{PS}_G(M)$ having the Zariski topology on the graded prime spectrum $Spec_G(M)$ as a subspace topology, and investigate several topological properties of this topological space.

math.AC↗

arXiv2021

The Quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. A proper graded submodule $Q$ of $M$ is called a graded quasi-primary submodule if whenever $r\in h(R)$ and $m\in h(M)$ with $rm\in Q$, then either $r\in Gr((Q:_{R}M))$ or $m\in Gr_{M}(Q)$. The graded quasi primary spectrum $qp.Spec_{g}(M)$ is defined to be the set of all graded quasi primary submodules of $M$. In this paper, we introduce and study a topology on $qp.Spec_{g}(M)$, called the Quasi-Zariski Topology, and investigate properties of this topology and some conditions under which (% $qp.Spec_{g}(M)$, $q.τ^{g}$) is a Noetherian, spctral space.

math.AC↗

arXiv2021

On Graded classical 2-absorbing second submodules of graded modules over graded commutative rings

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 classical and graded strongly classical 2-absorbing second submodules of graded modules over a graded commutative rings. A number of results concerning these classes of graded submodules and their homogeneous components are given.

math.AC↗

arXiv2021

On graded $I_{e}$-prime submodules of graded modules over graded commutative rings

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $I_{e}$-prime submodule as a generalization of a graded prime submodule for $I=\oplus_{g\in G}I_{g}$ a fixed graded ideal of $R$. We give a number of results concerning of these classes of graded submodules and their homogeneous components. A proper graded submodule $N$ of $M$ is said to be a graded $I_{e}$-prime submodule of $M$ if whenever $% r_{g}\in h(R)$ and $m_{h}\in h(M)$ with $r_{g}m_{h}\in N-I_{e}N,$ then either $r_{g}\in (N:_{R}M)$ or $m_{h}\in N.$

math.AC↗

arXiv2020

On graded $A$-2-absorbing submodules of graded modules over graded commutative rings

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring, $M$ a graded $R$-module and $A\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $A$-2-absorbing submodules of $M$ as a generalization of graded 2-absorbing submodules and graded $A$-prime submodules of $M.$ We investigate some properties of this class of graded submodules.

math.AC↗

arXiv2020

Some properties of graded generalized 2-absorbing submodules

Let $G$ be an abelian 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 generalized 2-absorbing submodules and their homogeneous components. Special attention has been paid, when graded rings are graded gr-Noetherian, to find extra properties of these graded submodules.

math.AC↗

arXiv2020

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

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 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.

math.AC↗