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Journal of Discrete Mathematical Sciences and Cryptography

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Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

On graded 1-absorbing prime submodules

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pp. 2167–2181Vol. 26Issue 8December 2023DOI: 10.1080/09720529.2022.2026623 Crossmark XML
Received:
01 May 2021
Accepted:
01 Aug 2021
Published Online:
22 Aug 2021
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-1457
Pages:
2167–2181

Abstract

Let G be a group with identity e, R be a commutative G-graded ring with unity 1 and M be a G-graded unital R-module. In this article, we introduce the concept of graded 1-absorbing prime submodule. A proper graded R-submodule N of M is said to be a graded 1-absorbing prime R-submodule of M if for all non-unit homogeneous elements x, y of R and homogeneous element m of M with xym ∈ N, either xy ∈ (N :R M) or m ∈ N. We show that the new concept is a generalization of graded prime submodules. At the same time it is a special graded 2-absorbing submodule. Several properties of a graded 1-absorbing prime submodule have been obtained. We investigate graded 1-absorbing prime submodules when the components {Mg : g ∈ G} are multiplication Re-modules.

Keywords

Graded 1-absorbing prime idealGraded 1-absorbing prime submoduleGraded prime submodule

Subject Classifications

(2020) Primary 16W50Secondary 13A02

References

[1] R. Abu-Dawwas, M. Bataineh and H. Shashan, Graded generalized 2-absorbing submodules, Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry, (2020), https://doi.org/10.1007/s13366-020-00544-1.
[2] R. Abu-Dawwas, E. Yldz, Ü. Tekir and S. Koç, On graded 1-absorbing prime ideals, Sao Paulo Journal of Mathematical Sciences, 15, 450-462 (2021). https://doi.org/10.1007/s40863-021-00218-3.
[3] K. Al-Zoubi and R. Abu-Dawwas, On graded 2-absorbing and weakly graded 2-absorbing submodules, Journal of Mathematical Sciences: Advances and Applications, 28 (2014), 45-60.
[4] K. Al-Zoubi, R. Abu-Dawwas and S. Çeken, On graded 2-absorbing and graded weakly 2-absorbing ideals, Hacettepe Journal of Mathematics and Statistics, 48 (3) (2019), 724-731.
[5] K. Al-Zoubi and F. Qarqaz, An intersection condition for graded prime ideals, Bollettino dell’Unione Matematica Italiana, 11 (2018), 483-488.
[6] S. E. Atani, On graded prime submodules, Chiang Mai Journal of Science, 33 (1) (2006), 3-7.
[7] S. E. Atani and F. E. K. Saraei, Graded modules which satisfy the gr-radical formula, Thai Journal of Mathematics, 8 (1) (2010), 161-170.
[8] A. Badawi and E. Y. Celikel, On 1-absorbing primary ideals of commutative rings, Journal of Algebra and Its Applications, 19 (6), 205-211 (2020).
[9] Z. A. El-Bast and P. F. Smith, Multiplication modules, Communications in Algebra, 16 (1988), 755-779.
[10] F. Farzalipour, P. Ghiasvand, On the union of graded prime submodules, Thai Journal of Mathematics, 9 (1) (2011), 49-55.
[11] M. Hamoda and A. E. Ashour, On graded n-ansorbing submodules, Le Matematiche, (2015), 243-254, doi:10.4418/2015.70.2.16.
[12] R. Hazrat, Graded rings and graded Grothendieck groups, Cambridge University press, 2016.
[13] S. R. Naghani and H. F. Moghimi, On graded 2-absorbing and graded weakly 2-absorbing ideals of a commutative ring, Çankaya University Journal of Science and Engineering, 13 (2) (2016), 11-17.
[14] C. Nastasescu and F. Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, 1836, Springer-Verlag, Berlin, 2004.
[15] M. Refai and K. Al-Zoubi, On graded primary ideals, Turkish Journal of Mathematics, 28 (2004), 217-229.
[16] M. Refai, M. Hailat and S. Obiedat, Graded radicals and graded prime spectra, Far East Journal of Mathematical Sciences, (2000), 59-73.
[17] E. A. Ugurlu, On 1-absorbing prime submodules, (2020), https://arxiv.org/abs/2007.01103, submitted.
[18] A. Yassine, M. J. Nikmehr, R. Nikandish, On 1-absorbing prime ideals of commutative rings, Journal of Algebra and its Applications, (2020), https://doi:10.1142/S0219498821501759.

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