Document Type : Original Manuscript

Authors

Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.

Abstract

Let $G$ be a group with identity $e$, $R$ be an associative graded ring and $M$ be a $G$-graded $R$-module. In this article, we intruduce the concept of graded semiprime
submodules over non-commutative graded rings. First, we study graded prime $R$-modules
over non-commutative graded rings and we get some properties of such graded modules.
Second, we study graded semiprime and graded radical submodules of graded $R$-modules.
For example, we give some equivalent conditions for a graded module to have zero graded

Keywords

###### ##### References
1. R. Abu-Dawwas and M. Bataineh, Graded prime submodules over noncommutative rings, Vietnam J. Math., 46(3) (2018), 681–692.
2. S. Ebrahimi Atani, On graded prime submodules, Chiang Mai J. Sci., 33(1) (2006), 3–7.
3. S. Ebrahimi Atani and F. Farzalipour, Notes on the graded prime submodules, Int. Math. Forum, 38(1) (2006), 1871–1880.
4. S. Ebrahimi Atani and F. Farzalipour, On graded secondary modules, Turk. J. Math., 31 (2007), 371–378.
5. S. Ebrahimi Atani and Ü. Tekir, On the graded primary avoidance theorem, Chiang Mai J. Sci., 34(2) (2007), 161–164.
6. F. Farzalipour and P. Ghiasvand, A generalization of graded prime submodules over non-commutative graded rings, J. Algebra and Related Topics, 8(1) (2020), 39–50.
7. F. Farzalipour and P. Ghiasvand, On graded semiprime and graded weakly semiprime ideals, Int. Electronic J. Algebra, 13 (2013), 15–22.
8. F. Farzalipour and P. Ghiasvand, On the union of graded prime submodules, Thai. J. Math., 9(1) (2011), 49–55.
9. F. Farzalipour, P. Ghiasvand and M. Adlifar, On graded weakly semiprime submodules, Thai. J. Math., 12(1) (2014), 167–174.
10. P. Ghiasvand and F. Farzalipour, On graded weak multiplication modules, Tamkang J. Math., 43(2) (2012), 171–177.