On graded primary-like submodules of graded modules over graded commutative rings

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-3467

Keywords:

Graded primary ideals, Graded primary-like submodules, Graded prime submodules, gr-primeful property

Abstract

Let G be a group with identity e. Let R be a G-graded commutative ring andM a graded R-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to and extra properties of these graded submodules.

Author Biographies

Khaldoun Al-Zoubi, Jordan University of Science and Technology.

Dept. of Mathematics and Statistics.

Mohammed Al-Dolat, Jordan University of Science and Technology.

Dept. of Mathematics and Statistics.

References

K. Al-Zoubi, “The graded primary radical of a graded submodules”, Analele ştiinţifice ale Universităţii "Al.I. Cuza" din Iaşi. Matematică (Online), vol. 1, no. 2, pp. 395-402, 2016. [On line]. Available: https://bit.ly/36f5e3s

K. Al-Zoubi and R. Abu-Dawwas, “On graded quasi-prime submodules”, Kyungpook mathematical journal, vol. 55, no. 2, pp. 259-266, 2015. http://dx.doi.org/10.5666/KMJ.2015.55.2.259

K. Al-Zoubi, R. Abu-Dawwas and I. Al-Ayyoub, “Graded semiprime submodules and graded semi-radical of graded submodules in graded modules”, Ricerche di matematica, vol. 66, no. 2, pp. 449-455, 2017. https://doi.org/10.1007/s11587-016-0312-x

K. Al-Zoubi and M. Al-Dolat, “On graded classical primary submodules”, Advances in pure and applied mathematics, vol. 7, no. 2, pp. 93-96, 2016. https://doi.org/10.1515/apam-2015-0021

K. Al-Zoubi, M. Jaradat and R. Abu-Dawwas, “On graded classical prime and graded prime submodules”, Bulletin of the Iranian Mathematical Society, vol. 41, no. 1, pp. 217-225, 2015. [On line]. Available: https://bit.ly/2TB9K9Y

K. Al-Zoubi and F. Qarqaz, “An intersection condition for graded prime submodules in Gr-multiplication modules”, Mathematicals reports, vol. 20, no. 3, pp. 329-336, 2018. [On line]. Available: https://bit.ly/2TL9KUC

S. E. Atani, “On graded prime submodules”, Chiang mai journal of science, vol. 33, no. 1, pp. 3-7, 2006. [On line]. Available: https://bit.ly/3wmHTYr

S. E. Atani and R. E. Atani, “Graded multiplication modules and the graded ideal θg(M)”, Turkish journal of mathematics, vol. 35, no. 1, pp. 1-9, 2009. https://doi.org/10.3906/mat-0901-22

S. E. Atani and F. Farzalipour, “Notes on the graded prime submodules”, International mathematical forum, vol. 1, no. 38, pp. 1871-1880, 2006. http://dx.doi.org/10.12988/imf.2006.06162

S. E. Atani and F. E. K. Saraei, “Graded modules which satisfy the Gr-radical formula”, Thai journal of mathematics , vol. 8, no. 1, pp. 161-170, 2010. [On line]. Available: https://bit.ly/3e8mZWL

S. E. Atani and F. Farzalipour, “On graded secondary modules”, Turkish journal of mathematics, vol. 31, no. 4, pp. 371-378, 2007. [On line]. Available: https://bit.ly/3yt7wbh

J. Escoriza and B. Torrecillas, “Multiplication objects in commutative Grothendieck categories”, Communications in algebra, vol. 26, no. 6, pp. 1867-1883, 1998. https://doi.org/10.1080/00927879808826244

P. Ghiasvand and F. Farzalipour, “On graded primary submodules of graded multiplication modules”, International journal of algebra, vol. 4, no. 9, pp. 429-434, 2010. [On line]. Available: https://bit.ly/3wgHirb

R. Hazrat, Graded rings and graded Grothendieck groups, Cambridge: Cambridge University Press, 2016. https://doi.org/10.1017/CBO9781316717134

C. P. Lu, “A module whose prime spectrum has the surjective natural map”, Houston journal of mathematics, vol. 33, no. 1, pp. 125-143, 2007. [On line]. Available: https://bit.ly/36f7F68

S.C. Lee and R. Varmazyar, “Semiprime submodules of graded multiplication modules”, Journal of the Korean Mathematical Society, vol. 49, no. 2, pp. 435-447, 2012. https://doi.org/10.4134/JKMS.2012.49.2.435

H. F. Moghimi and F. Rashedi, “Primary-like submodules satisfying the primeful property”, Transactions on Algebra and its Applications, vol. 1, pp. 43-54, 2015.

C. Nastasescu and .F. Oystaeyen, Graded and filtered rings and modules. Berlin: Springer, 1979. https://doi.org/10.1007/BFb0067331

C. Nastasescu, F. Oystaeyen, Graded ring theory, Amsterdam: North Holland, 1982.

C. Nastasescu and F. Oystaeyen, Methods of graded rings. Berlin: Springer, 2004. https://doi.org/10.1007/b94904

K. H. Oral, U. Tekir and A.G. Agargun, “On Graded prime and primary submodules”, Turkish journal of mathematics, vol. 35, no. 2, pp. 159-167, 2011. https://doi.org/10.3906/MAT-0904-11

M. Refai and K. Al-Zoubi, “On graded primary ideals”, Turkish journal of mathematics, vol. 28, no. 3, pp. 217-229, 2004. [On line]. Available: https://bit.ly/3hMUvTs

M. Refai, M. Hailat and S. Obiedat, “Graded radicals on graded prime spectra”, Far East journal of mathematical sciences, vol. 1, no. 1, pp. 59-73, 2000.

H. A. Tavallaee and M. Zolfaghari, “Graded weakly semiprime submodules of graded multiplication modules”, Lobachevskii journal of mathematics, vol. 34, no. 1, pp. 61-67, 2013. https://doi.org/10.1134/S1995080213010113

Published

2021-04-19 — Updated on 2021-07-26

How to Cite

[1]
K. Al-Zoubi and M. Al-Dolat, “On graded primary-like submodules of graded modules over graded commutative rings”, Proyecciones (Antofagasta, On line), vol. 40, no. 4, pp. 859-871, Jul. 2021.

Issue

Vol. 40 No. 4 (2021)

Section

Artículos

Copyright (c) 2021 Khaldoun Falah Al-Zoubi, Mohammed Al-Dolat

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