On graded $J_{gr}$-classical prime submodules

Document Type : Research Paper


Department of Mathematics and Statistics, Faculty of Science and Arts Jordan University of Science and Technology, P.O.Box 3030, Irbid 22110, Jordan.



Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity 1 and $M$ a graded $R$-module. A proper graded submodule $C$ of $M$ is called a graded classical prime submodule if whenever $r,s\in h(R)$ and $m\in h(M)$ with $rsm\in C$, then either $rm\in C$ or $sm\in C$. In this paper, we introduce the concept of graded $J_{gr}$-classical prime submodule as a new generalization of graded classical submodule and we give some results concerning such graded modules. We say that a proper graded submodule $N$ of $M$ is \textit{a graded }$J_{gr}$\textit{-classical prime submodule of \ }$M$ if whenever $rsm\in N$ where $r,s\in h(R)$ and $m\in h(M)$, then either $rm\in N+J_{gr}(M)$ or $sm\in N+J_{gr}(M)$, where $J_{gr}(M)$ is the graded Jacobson radical.


[1] K. Al-Zoubi and M. Al-Dolat, On graded classical primary submodules, Adv. Pure Appl. Math., 7 No. 2 (2016) 93-96.
[2] K. Al-Zoubi and A. Al-Qderat, Some properties of graded comultiplication modules, Open Mathematics, 15 (2017) 187-192.
[3] K. Al-Zoubi and F. AL-Turman, On graded weakly classical primary submodules, Proc. Jangjeon Math. Soc., 21 No. 3 (2018) 405-412.
[4] K. Al-Zoubi and F. AL-Turman, On graded 2-classical prime submodules of graded modules over graded commutative rings, Proyecciones-Journal of Mathematics, 38 No. 4 (2019) 781-790.
[5] K. Al-Zoubi and S. Alghueiri, On graded Jgr-semiprime submodules, Ital. J. Pure Appl. Math, to appear.
[6] K. Al-Zoubi, M. Jaradat and R. Abu-Dawwa, On graded classical prime and graded prime submodules, Bull. Iranian Math. Soc., 41 No. 1 (2015) 217-225.
 [7] K. Al-Zoubi and M. Jaradat, The Zariski Topology on the graded classical prime spectrum of a graded module over a graded commutative ring, Matemati_cki Vesnik, 70 No. 4 (2018) 303-313.
[8] S. E. Atani, On graded prime submodules, Chiang Mai. J. Sci., 33 No. 1 (2006) 3-7.
[9] A. Y. Darani and S. Motmaen, Zariski topology on the spectrum of graded classical prime submodules, Appl. Gen. Topol., 14 No. 2 (2013) 159-169.
[10] R. Hazrat, Graded Rings and Graded Grothendieck Groups, Cambridge University Press, Cambridge, 2016.
[11] S.C Lee, R. Varmazyar, Semiprime submodules of Graded multiplication modules, J. Korean Math. Soc., 49 No. 2 (2012) 435-447.
[12] C. Nastasescu and F. Van Oystaeyen, Graded and _ltered rings and modules, Lecture notes in mathematics 758, Springer-Verlag, Berlin-New York, 1982.
[13] C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory, Mathematical Library 28, Amsterdam, North Holand, 1982.
[14] C. Nastasescu and F. Van Oystaeyen, Methods of Graded Rings, LNM 1836, Springer-Verlag, Berlin-Heidelberg, 2004.
[15] M. Refai and K. Al-Zoubi, On graded primary ideals, Turk. J. Math., 28 No. 3 (2004) 217-229.