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Graded Essential Extensions and Graded Injective Modules

Received: 20 January 2015     Accepted: 6 February 2015     Published: 11 February 2015
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Abstract

In this paper we establish the relation between graded essential extensions of graded modules and injectivity of such modules. We relate the graded injective hull Egr (M) of a graded module M with the graded essential extensions of M. We round off by establishing necessary and sufficient conditions for indecomposability of graded injective modules in terms of their graded injective hulls.

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 2)
DOI 10.11648/j.pamj.20150402.13
Page(s) 47-51
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Graded Injective Module, Graded Essential Extension, Graded Injective Hull

References
[1] Amin I., Yousif M., Zeyada N., Soc-injective rings and modules, Comm. in Alg., Vol. 33,No.1, (2005), 4229-4250.
[2] Zeyada Nasr A., Hussein Salah El Din S., Amin Amr K., Rad-injective and almost-injective modules and rings, Alg. Colloquium, Vol.18, No. 3, (2011), 411-418.
[3] Nicholson W.K., Yousif M.F., Quasi-Frobenius rings, Cambridge Tracts in Math. 158, Cambridge Univ. Press, Cambridge, UK, (2003).
[4] Yousif M. F., Zhou Y., Zeyada N., On peseudo-Frobenius rings, Can. Math. Bull., Vol.48 No. 2, (2005), 317-320.
[5] Nastasescu C., Van Oystaeyen F., Graded ring theory, Math. Library, North Holland, Amsterdam 28, (1982).
[6] Nastasescu C., Van Oystaeyen F., Methods of graded rings, Springer-Verlag, Berlin Heidelberg, LNM.1836, (2004).
[7] Kasch F., Modules and rings, Academic Press, New York, L.M.S. Monograph No.17, (1982).
[8] Lam T. Y., Lectures on modules and rings, Graduate Texts in Mathematics, Springer-Verlag, New York, Vol. 189, (1998).
[9] Papp, Zolt’an, On algebraically closed modules, Publicationes Mathematicae Debrecen, Vol.6, (1959), 311-327.
Cite This Article
  • APA Style

    Salah El Din S. Hussein, Essam El Seidy, H. S. Diab. (2015). Graded Essential Extensions and Graded Injective Modules. Pure and Applied Mathematics Journal, 4(2), 47-51. https://doi.org/10.11648/j.pamj.20150402.13

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    ACS Style

    Salah El Din S. Hussein; Essam El Seidy; H. S. Diab. Graded Essential Extensions and Graded Injective Modules. Pure Appl. Math. J. 2015, 4(2), 47-51. doi: 10.11648/j.pamj.20150402.13

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    AMA Style

    Salah El Din S. Hussein, Essam El Seidy, H. S. Diab. Graded Essential Extensions and Graded Injective Modules. Pure Appl Math J. 2015;4(2):47-51. doi: 10.11648/j.pamj.20150402.13

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  • @article{10.11648/j.pamj.20150402.13,
      author = {Salah El Din S. Hussein and Essam El Seidy and H. S. Diab},
      title = {Graded Essential Extensions and Graded Injective Modules},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {2},
      pages = {47-51},
      doi = {10.11648/j.pamj.20150402.13},
      url = {https://doi.org/10.11648/j.pamj.20150402.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150402.13},
      abstract = {In this paper we establish the relation between graded essential extensions of graded modules and injectivity of such modules. We relate the graded injective hull Egr (M) of a graded module M with the graded essential extensions of M. We round off by establishing necessary and sufficient conditions for indecomposability of graded injective modules in terms of their graded injective hulls.},
     year = {2015}
    }
    

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    T1  - Graded Essential Extensions and Graded Injective Modules
    AU  - Salah El Din S. Hussein
    AU  - Essam El Seidy
    AU  - H. S. Diab
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    DO  - 10.11648/j.pamj.20150402.13
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    EP  - 51
    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.pamj.20150402.13
    AB  - In this paper we establish the relation between graded essential extensions of graded modules and injectivity of such modules. We relate the graded injective hull Egr (M) of a graded module M with the graded essential extensions of M. We round off by establishing necessary and sufficient conditions for indecomposability of graded injective modules in terms of their graded injective hulls.
    VL  - 4
    IS  - 2
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Author Information
  • Department of Mathematics, Faculty of Science, Ain Shams University, Abasaya-Cairo, Egypt

  • Department of Mathematics, Faculty of Science, Ain Shams University, Abasaya-Cairo, Egypt

  • Department of Mathematics, Faculty of Science, Ain Shams University, Abasaya-Cairo, Egypt

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