Volume 25 • Issue 3 • PP: 265-279 • 2025
The Zariski topology on the graded second spectrum of a graded module
Open Access & Copyright
© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
View PDF Abstract
Let R be a G-graded ring and M be a G-graded R-module. The graded second spectrum of M, denoted by Specs G(M), is the set of all graded second submodules of M. In this paper, we define a topology on Specs G(M) which is analogous to that for SpecG(R), and investigate several topological properties of this topology.
Keywords
References
[1] K. Al-Zoubi and M and Al-Azaizeh, On graded 2-absorbing second submodules of graded modules over graded commutative rings, Kragujevac Journal of Mathematics 48 (1) (2024), 55-66.
[2] K. Al-Zoubi and A. Al-Qderat, Some properties of graded comultiplication modules, Open Math., 15 (1) (2017), 187-192.
[3] K. Al-Zoubi and M. Jaradat, The Zariski Topology on the graded classical prime spectrum of a graded module over a graded commutative ring, Mat. Vesnik, 70 (2018), 303-313.
[4] K. Al-Zoubi and M. Jaradat,The Zariski topology on the graded primary spectrum over graded commutative rings, Tatra Mt. Math. Publ. 74 (1) (2019), 7-16.
[5] H. Ansari-Toroghy and F.Farshadifar, On graded second modules, Tamkang J. Math., 43 (4) (2012), 499-505
[6] H. Ansari-Toroghy and F. Farshadifar, Graded comultiplication modules, Chiang Mai J. Sci., 38 (3) (2011), 311-320.
[7] N. Bourbaki, Commutative Algebra. Chapter 1–7. Springer-Verlag, Berlin (1989).
[8] S. C¸ eken and M. Alkan, On graded second and coprimary modules and graded secondary representations, Bull. Malays. Math. Sci. Soc., 38 (4) (2015), 1317-1330.
[9] C. Nastasescu and F. Van Oystaeyen, Methods of Graded Rings, LNM 1836. Berlin-Heidelberg: Springer-Verlag, 2004.
[10] N. A. Ozkirisci , K.H. Oral, U. Tekir, Graded prime spectrum of a graded module, Iran. J. Sci.Technol., 37 A3 (2013), 411-420.
[11] M. Refai, M. Hailat and S. Obiedat, Graded radicals on graded prime spectra, Far East J. of Math. Sci., part I (2000), 59-73.
[12] M. Refai, On properties of G-spec(R), Sci. Math. Jpn. 53 (3) (2001), 411-415.
[13] M. Refai and R.Abu-Dawwas, On generalizations of graded second submodules, Proyecciones (Antofagasta) 39 (6) (2020), 1537-1554.
[14] S. Salam and K. Al-Zoubi, The Zariski topology on the graded primary spectrum of a graded module over a graded commutative ring, Applied General Topology 23(2) (2022), 345-361.
[15] F. Van Oystaeyen, Methods in ring theory. Vol. 129. Springer Science and Business Media, 2012
Cite This Article
Choose your preferred format
Publisher's Note
The statements, opinions, and data presented in this article are solely those of the author(s) and do not necessarily represent those of ASPG, the journal, or its editors. ASPG and the editors disclaim responsibility for any harm arising from the use of any ideas, methods, instructions, or products described in this article, to the fullest extent permitted by applicable law.