Shahrood University of Technology Journal of Algebraic Systems 2345-5128 10 1 2022 09 01 GRADED SEMIPRIME SUBMODULES OVER NON-COMMUTATIVE GRADED RINGS 95 110 2324 10.22044/jas.2021.9102.1442 EN P. Ghiasvand Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran. F. Farzalipour Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran. Journal Article 2019 11 11 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<br />submodules over non-commutative graded rings. First, we study graded prime $R$-modules<br />over non-commutative graded rings and we get some properties of such graded modules.<br />Second, we study graded semiprime and graded radical submodules of graded $R$-modules.<br />For example, we give some equivalent conditions for a graded module to have zero graded<br />radical submodule. https://jas.shahroodut.ac.ir/article_2324_0c4acb774102cb642394ff5b9a1d337a.pdf