## Definition

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

Let $M$ be an $R$-module.

We say $M$ is $\N$-graded if and only if it has a decomposition as a direct sum of abelian groups:

$\ds R = \bigoplus_{n \mathop \in \N} R_n$

such that for all $m, n \in \N$ we have $R_n M_m \subseteq M_{n + m}$.

Similarly, we say $M$ is $\Z$-graded if and only if it has a decomposition as a direct sum of abelian groups:

$\ds R = \bigoplus_{n \mathop \in \Z} R_n$

such that for all $m, n \in \Z$ we have $R_n M_m \subseteq M_{n + m}$.