Prime Submodules of Graded Modules

Authors

  • Rashid Abu-Dawwas Yarmouk University
  • Khaldoun Al-Zoubi Jordan University Of Science And Technology.
  • Malik Bataineh Jordan University Of Science And Technology.

DOI:

https://doi.org/10.4067/S0716-09172012000400004

Keywords:

Graded rings, graded modules, prime submodules, álgebra graduada, módulos graduados, submódulos primos.

Abstract

Let G be a group, R be a G-graded ring and M be a G-graded R-module. Suppose P is a prime ideal of Reand g G G. In this article, we define

Mg(P) = {m G Mg: Am C PMg for some ideal A of Re satisfying A C P}

that is an Re-submodule of Mg, and we investigate some results on this submodule. Also, we introduce a situation where if N is a gr-prime R-submodule of M ,then (Ng: Mg) is a maximal ideal of Re. We close this article by introducing a situation where if N is a gr- R-submodule ofM such that Ne is a weakly prime Re-submodule ofMe,thenNg is a prime Re-submodule of Mg.

Author Biographies

Rashid Abu-Dawwas, Yarmouk University

Department of Mathematics.

Khaldoun Al-Zoubi, Jordan University Of Science And Technology.

Department of Mathematics and Statistics.

Malik Bataineh, Jordan University Of Science And Technology.

Department of Mathematics and Statistics.

References

[1] R. Abu-Dawwas, More on crossed product over the support of graded rings, International Mathematical Forum, 5 (63) (2010), 3121 − 3126.

[2] R. Abu-Dawwas and M. Refai, Further results on graded prime submodules, International Journal of Algebra, 4 (28) (2010), 1413 − 1419.

[3] E. C. Dade, Group graded rings and modules, Math. Z., 174 (1980), 241 − 262.

[4] C. Nastasescu, Strongly graded rings of finite groups, Comm. Algebra, 11 (10) (1981), 1033 − 1071.

[5] C. Nastasescu and F. Van Oystaeyen, Graded ring theory, Mathematical Library 28, North Holland, Amesterdam, 1982.

[6] M. Refai, Various types of strongly graded rings, Abhath Al-Yarmouk Journal (Pure Sciences and Engineering Series), 4 (2) (1995), 9 − 19.

[7] M. Refai and K. Al-Zoubi, On graded primary ideals, Turkish J. Mathematics, 28 (2004), 217 − 229.

Published

2013-02-19

How to Cite

[1]
R. Abu-Dawwas, K. Al-Zoubi, and M. Bataineh, “Prime Submodules of Graded Modules”, Proyecciones (Antofagasta, On line), vol. 31, no. 4, pp. 355-361, Feb. 2013.

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