[[Graded module]]
# Graded submodule

Let $M$ be a [[graded module]] over a $\mathfrak{A}$-[[graded ring]].
A [[submodule]] $N \leq M$ is said to be graded iff #m/def/module 
$$
\begin{align*}
N &= \bigoplus _{\alpha \in \mathfrak{A}} N_{\alpha}
\end{align*}
$$
where $N_{\alpha} = N \cap M_{\alpha}$ for $\alpha \in \mathfrak{A}$.
In this case the inclusion $\iota: N \hookrightarrow M$ is a [[homomorphism of graded modules]]
and one may construct the [[quotient graded module]].

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#state/tidy | #lang/en | #SemBr