# Definition:Graded Module

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## 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 $R_n M_m \subseteq M_{n + m}$.