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AIMS Mathematics

2020, Volume 5, Issue 6: 7624-7631. doi: 10.3934/math.2020487
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Research article

On graded 2-absorbing $I_{e}$-prime submodules of graded modules over graded commutative rings

  • Department of Mathematics and Statistics, Jordan University of Science and Technology, P. O. Box 3030, Irbid 22110, Jordan
  • Received: 11 July 2020 Accepted: 22 September 2020 Published: 25 September 2020

  • MSC : 13A02, 16W50

  • Let $G$ be an abelian 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 2-absorbing $I_{e}$-prime submodule as a generalization of a graded 2-absorbing prime submodule for $\ I = \oplus _{g\in G}I_{g}$ a fixed graded ideal of $R$. We give a number of results concerning these classes of graded submodules and their homogeneous components. A proper graded submodule $N$ of $M$ is said to be a graded 2-absorbing $I_{e}$-prime submodule of $M$ if whenever $% r_{h}, s_{\lambda }\in h(R)$ and $m_{\alpha }\in h(M)$ with $r_{h}s_{\lambda }m_{\alpha }\in N\backslash I_{e}N$, implies either $r_{h}s_{\lambda }\in (N:_{R}M)$ or $r_{h}m_{\alpha }\in N$ or $s_{\lambda }m_{\alpha }\in N.$

    Citation: Shatha Alghueiri, Khaldoun Al-Zoubi. On graded 2-absorbing $I_{e}$-prime submodules of graded modules over graded commutative rings[J]. AIMS Mathematics, 2020, 5(6): 7624-7631. doi: 10.3934/math.2020487

    Related Papers:

  • Let $G$ be an abelian 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 2-absorbing $I_{e}$-prime submodule as a generalization of a graded 2-absorbing prime submodule for $\ I = \oplus _{g\in G}I_{g}$ a fixed graded ideal of $R$. We give a number of results concerning these classes of graded submodules and their homogeneous components. A proper graded submodule $N$ of $M$ is said to be a graded 2-absorbing $I_{e}$-prime submodule of $M$ if whenever $% r_{h}, s_{\lambda }\in h(R)$ and $m_{\alpha }\in h(M)$ with $r_{h}s_{\lambda }m_{\alpha }\in N\backslash I_{e}N$, implies either $r_{h}s_{\lambda }\in (N:_{R}M)$ or $r_{h}m_{\alpha }\in N$ or $s_{\lambda }m_{\alpha }\in N.$


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