Graded Essential Extensions and Graded Injective Modules

Salah El Din S. Hussein; Essam El Seidy; H. S. Diab

doi:doi:10.11648/j.pamj.20150402.13

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Graded Essential Extensions and Graded Injective Modules

Salah El Din S. Hussein, Essam El Seidy, H. S. Diab

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 2)

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.

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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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TY  - JOUR
T1  - Graded Essential Extensions and Graded Injective Modules
AU  - Salah El Din S. Hussein
AU  - Essam El Seidy
AU  - H. S. Diab
Y1  - 2015/02/11
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JO  - Pure and Applied Mathematics Journal
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VL  - 4
IS  - 2
ER  -

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Author Information

Salah El Din S. Hussein

Department of Mathematics, Faculty of Science, Ain Shams University, Abasaya-Cairo, Egypt
Essam El Seidy

Department of Mathematics, Faculty of Science, Ain Shams University, Abasaya-Cairo, Egypt
H. S. Diab

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

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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}
}

Copy | Download

TY  - JOUR
T1  - Graded Essential Extensions and Graded Injective Modules
AU  - Salah El Din S. Hussein
AU  - Essam El Seidy
AU  - H. S. Diab
Y1  - 2015/02/11
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150402.13
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
SP  - 47
EP  - 51
PB  - Science Publishing Group
SN  - 2326-9812
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
ER  -

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