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.
DOI | 10.11648/j.pamj.20150402.13 |
Published in | Pure and Applied Mathematics Journal (Volume 4, Issue 2, April 2015) |
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), 2024. Published by Science Publishing Group |
Graded Injective Module, Graded Essential Extension, Graded Injective Hull
[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. |
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
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
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
@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://download.sciencepg.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} }
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 -