Research article

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

  • 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 $s_{\lambda }m_{\alpha }\in N.$


    加载中


    [1] K. Al-Zoubi, Some properties of graded 2-prime submodules, Asian-European Journal of Mathematics, 8 (2015), 1550016.
    [2] K. Al-Zoubi, R. Abu-Dawwas, On graded quasi-prime submodules, Kyungpook Mathematical Journal, 55 (2015), 259-266.
    [3] K. Al-Zoubi, R. Abu-Dawwas, On graded 2-absorbing and weakly graded 2-absorbing submodules, Journal of mathematical sciences: advances and applications, 28 (2014), 45-60.
    [4] K. Al-Zoubi, R. Abu-Dawwas, S. Çeken, On graded 2-absorbing and graded weakly 2-absorbing ideals, Hacet. J. Math. Stat., 48 (2019), 724-731.
    [5] K. Al-Zoubi, M. Al-Azaizeh, On graded classical 2-absorbing submodules of graded modules over graded commutative rings, Rend. Istit. Mat. Univ. Trieste, 50 (2018), 37-46.
    [6] K. Al-Zoubi, M. Al-Azaizeh, On graded weakly 2-absorbing primary submodules, Vietnam J. Math., 47 (2019), 297-307.
    [7] K. Al-Zoubi, M. Al-Azaizeh, Some properties of graded 2-absorbing and graded weakly 2- absorbing submodules, J. Nonlinear Sci. Appl., 12 (2019), 503-508.
    [8] K. Al-Zoubi, I. Al-Ayyoub, M. Al-Dolat, On graded 2-absorbing compactly packed modules, Adv. Stud. Contemp. Math. (Kyungshang), 28 (2018), 479-486.
    [9] K. Al-Zoubi, A. Al-Qderat, Some properties of graded comultiplication modules, Open Math., 15 (2017), 187-192.
    [10] K. Al-Zoubi, M. Jaradat, R. Abu-Dawwas, On graded classical prime and graded prime submodules, B. Iran. Math. Soc., 41 (2015), 217-225.
    [11] K. Al-Zoubi, F. Qarqaz, An Intersection condition for graded prime submodules in Grmultiplication modules, Math. Rep., 20 (2018), 329-336.
    [12] K. Al-Zoubi, B. Rabab'a, Some properties of graded prime and graded weakly prime submodules, FJMS., 102 (2017), 1613-1624.
    [13] S. Alghueiri, K. Al-Zoubi, On graded Ie-prime submodules of graded modules over graded commutative rings, Preprint.
    [14] S. E. Atani, On graded prime submodules, Chiang Mai J. Sci., 33 (2006), 3-7.
    [15] A. Badawi, On 2-absorbing ideals of commutative rings, B. Aust. Math. Soc., 75 (2007), 417-429.
    [16] E. Y. Celikel, On graded 2-absorbing primary submodules, International Journal of Pure and Applied Mathematics, 109 (2016), 869-879.
    [17] A. Y. Darani, F. Soheilnia, 2-absorbing and weakly 2-absorbing submoduels, Thai Journal of Mathematics, 9 (2012), 577-584.
    [18] F. Farshadifar, 2-absorbing I-prime and 2-absorbing I-second submodules, Algebraic Structures and Their Applications, 6 (2019), 47-55.
    [19] R. Hazrat, Graded Rings and Graded Grothendieck Groups, Cambridge University Press, 2016.
    [20] C. Nastasescu, F. Van Oystaeyen, Graded and filtered rings and modules, Springer-Verlag, Berlin Heidelberg, 1979.
    [21] C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory, Mathematical Library, 1982.
    [22] C. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, Springer, 2004.
    [23] K. H. Oral, Ü. Tekir, A. G. Ağargün, On graded prime and primary submodules, Turk. J. Math., 35 (2011), 159-167.
    [24] S. Payrovi, S. Babaei, On 2-absorbing submodules, Algebr. Colloq., 19 (2012), 913-920.
    [25] M. Refai, K. Al-Zoubi, On graded primary ideals, Turk. J. Math., 28 (2004), 217-230.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2699) PDF downloads(99) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog