Volume 27 • Issue 2 • PP: 297-302 • 2026
On Graded Weakly Jgr-classical Prime Submodules
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© 2026 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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Let 0 be a group, Υ be a 0-graded commutative ring with unity 1 and M a graded Υ-module. Our goal in this paper, introducing the concept of graded weakly Jgr -classical prime submodule as a generalization of graded weakly classical prime submodule and offering several results pertinent of graded weakly Jgr - classical prime submodules. For instance, we give characterizations of graded weakly Jgr -classical prime submodule. Also, we give some restrictions for graded submodule to be a graded weakly Jgr -classical prime submodule. A proper graded submodule V of M is said to be a graded weakly Jgr -classical prime submodule of M if, whenever 0̸ = abx ∈ V where a, b ∈ h(Υ) and x ∈ h(M), then either ax ∈ V + Jgr (M) or bx ∈ V + Jgr (M), The symbol Jgr (M) indicates the graded Jacobson radical of Υ-module M.
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References
[1] C. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, LNM, 1836, Berlin-Heidelberg: Springer- Verlag, 2004.
[2] S.E. Atani, On graded prime submodules, Chiang Mai. J. Sci., 33(1) (2006), 3-7.
[3] K. H. Oral, U. Tekir, A. G. Agargun, On graded prime and primary submodules, Turk. J. Math. 35(2) (2011), 159–167.
[4] Atani S. E. On graded weakly prime submodules, In Int. Math. Forum 1(2) (2006): 61-66.
[5] Motmaen S. On graded classical prime submodules, In Second Seminar on Algebra and its Applications: 83-86.
[6] A.Y. Darani and S. Motmaen. Zariski topology on the spectrum of graded classical prime submodules. Applied general topology, 14(2) (2013), 159-169.
[7] R. Abu-Dawwas and K. Al-Zoubi. On graded weakly classical prime submodules. Iranian Journal of Mathematical Sciences and Informatics 1(12)(2017), 153-161.
[8] Jebrel H. and R. Abu-Dawwas. Graded classical weakly prime submodules over non-commutative graded rings. Tatra Mt. Math. Publ, 87 (2024), 85-104.
[9] H. Ansari-Toroghy and F. Farshadifar. Graded comultiplication modules. Chiang mai journal of science, 38(3) (2011), 311-320.
[10] K. Al-Zoubi and S. Alghueiri. On graded Jgr -semiprime submodules. Ital. J. Pure Appl. Math. 46 (2021), 361-369.
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