Document Type : Original Manuscript


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


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
radical submodule.


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