Definition:Graded Submodule/Definition 2

Definition
Let $G \in \set {\N, \Z}$.

Let $R$ be a $G$-graded commutative ring with unity.

Let $M = \bigoplus_{n \in G} M_n$ be a $G$-graded $R$-module.

Let $N$ be a submodule of $M$.

$N$ is $G$-graded $N$ is generated over $R$ by homogeneous elements of $M$.