Definition:Graded Submodule/Definition 3

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

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

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

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

$N$ is graded :
 * $x_r + x_{r+1} + \cdots + x_s \in N$ such that $\forall i : x_i \in M_i$
 * $\implies \forall i : x_i \in N$

Also see

 * Equivalence of Definitions of Graded Submodule