\begin{definition}[Definition:Graded Module]
Let $R$ be a graded commutative ring with unity.
Let $M$ be an $R$-module.
We say $M$ is '''$\N$-graded''' {{iff}} 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''' {{iff}} 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 $R_n M_m \subseteq M_{n + m}$.
Category:Definitions/Module Theory
\end{definition}