A Note on Localization of Supplemented Modules

Esra Öztürk; Şenol Eren

doi:doi:10.11648/j.pamj.20140303.12

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A Note on Localization of Supplemented Modules

Esra Öztürk, Şenol Eren

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 3)

Received: 4 March 2014 Accepted: 27 June 2014 Published: 10 July 2014

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Abstract

In this paper we study on commutative rings with identity and all modules are unital left R-modules unless otherwise stated. We define the concept of small submodules for localization modules and additionally, we present the relation between an R-module M and an R_p-localization module M_p for all maximal ideals of R in view of being supplemented.

Published in	Pure and Applied Mathematics Journal (Volume 3, Issue 3)
DOI	10.11648/j.pamj.20140303.12
Page(s)	66-69
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

Keywords

Small Submodule, Supplemented Module, Multiplicative Closed Set, Localization Module

References

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[2]	Anderson, F.W., Fuller K.R., 1974, Rings and Categories of Modules, Springer-Verlag, New York-Heidelberg-Berlin.
[3]	Çallıalp, F., Tekir, Ü., 2009. Değişmeli Halkalar ve Modüller, Birsen Yayınevi, İstanbul.
[4]	Faith, C., 1976, Algebra II: Ring Theory, Springer- Verlag, Berlin.
[5]	Hungerford, T.W., 1973. Algebra, Springer-Verlag, New York.
[6]	Lam, T. Y., Lectures on modules and rings, ser. Graduate texts in Mathematics. New York: Springer, 1999, vol. 189.
[7]	Larsen, M.D., McCarthy, P.J., 1971. Multiplicative Theory of Ideals, Academic Press, New York and London.
[8]	Moore, D.G., 1968. Prime and Radical Submodules of Modules over Commutative Rings, Vol.30, No.10, 5037-5064.
[9]	Mohamed, S. H. and Müller, B. J., Continuous and discrete modules, ser. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1990, vol. 147.
[10]	Sharpe, D.W., Vamos, P., 1972. Injective Modules, Cambridge University Press.
[11]	Taşçı, D., 2007. Soyut Cebir, Alp Yayınevi, Ankara.
[12]	Northcott, D.G., 1968. Lessons on Rings, Modules and Multiplicity, Cambridge University Press.
[13]	J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, 2006. Lifting Modules, Supplements and projectivity in module theory, Front. Math., Birkhauser Verlag, Basel.
[14]	Wisbauer, R., 1991. Foundations of Module and Ring Theory, Gordon and Breach (Philadelphia).
[15]	Shang, Y. On the ideals of commutative local rings, Kochi Journal of Mathematics, 2013, vol. 8, 13--17.

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APA Style

Esra Öztürk, Şenol Eren. (2014). A Note on Localization of Supplemented Modules. Pure and Applied Mathematics Journal, 3(3), 66-69. https://doi.org/10.11648/j.pamj.20140303.12

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ACS Style

Esra Öztürk; Şenol Eren. A Note on Localization of Supplemented Modules. Pure Appl. Math. J. 2014, 3(3), 66-69. doi: 10.11648/j.pamj.20140303.12

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AMA Style

Esra Öztürk, Şenol Eren. A Note on Localization of Supplemented Modules. Pure Appl Math J. 2014;3(3):66-69. doi: 10.11648/j.pamj.20140303.12

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@article{10.11648/j.pamj.20140303.12,
  author = {Esra Öztürk and Şenol Eren},
  title = {A Note on Localization of Supplemented Modules},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {3},
  pages = {66-69},
  doi = {10.11648/j.pamj.20140303.12},
  url = {https://doi.org/10.11648/j.pamj.20140303.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140303.12},
  abstract = {In this paper we study on commutative rings with identity and all modules are unital left R-modules unless otherwise stated. We define the concept of small submodules for localization modules and additionally, we present the relation between an R-module M and an R_p-localization module M_p for all maximal ideals of R in view of being supplemented.},
 year = {2014}
}

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TY  - JOUR
T1  - A Note on Localization of Supplemented Modules
AU  - Esra Öztürk
AU  - Şenol Eren
Y1  - 2014/07/10
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140303.12
DO  - 10.11648/j.pamj.20140303.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 66
EP  - 69
PB  - Science Publishing Group
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AB  - In this paper we study on commutative rings with identity and all modules are unital left R-modules unless otherwise stated. We define the concept of small submodules for localization modules and additionally, we present the relation between an R-module M and an R_p-localization module M_p for all maximal ideals of R in view of being supplemented.
VL  - 3
IS  - 3
ER  -

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Esra Öztürk
Şenol Eren

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APA Style

Esra Öztürk, Şenol Eren. (2014). A Note on Localization of Supplemented Modules. Pure and Applied Mathematics Journal, 3(3), 66-69. https://doi.org/10.11648/j.pamj.20140303.12

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ACS Style

Esra Öztürk; Şenol Eren. A Note on Localization of Supplemented Modules. Pure Appl. Math. J. 2014, 3(3), 66-69. doi: 10.11648/j.pamj.20140303.12

Copy | Download

AMA Style

Esra Öztürk, Şenol Eren. A Note on Localization of Supplemented Modules. Pure Appl Math J. 2014;3(3):66-69. doi: 10.11648/j.pamj.20140303.12

Copy | Download

@article{10.11648/j.pamj.20140303.12,
  author = {Esra Öztürk and Şenol Eren},
  title = {A Note on Localization of Supplemented Modules},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {3},
  pages = {66-69},
  doi = {10.11648/j.pamj.20140303.12},
  url = {https://doi.org/10.11648/j.pamj.20140303.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140303.12},
  abstract = {In this paper we study on commutative rings with identity and all modules are unital left R-modules unless otherwise stated. We define the concept of small submodules for localization modules and additionally, we present the relation between an R-module M and an R_p-localization module M_p for all maximal ideals of R in view of being supplemented.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - A Note on Localization of Supplemented Modules
AU  - Esra Öztürk
AU  - Şenol Eren
Y1  - 2014/07/10
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140303.12
DO  - 10.11648/j.pamj.20140303.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 66
EP  - 69
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140303.12
AB  - In this paper we study on commutative rings with identity and all modules are unital left R-modules unless otherwise stated. We define the concept of small submodules for localization modules and additionally, we present the relation between an R-module M and an R_p-localization module M_p for all maximal ideals of R in view of being supplemented.
VL  - 3
IS  - 3
ER  -

Copy | Download