Let $R$ be a graded ring and $M$ a graded$R$-module. $M$ is called a graded free module if $M$ has a Basis consisting of homogeneous elements
Now in my lecture note there is the following Theorem without a proof.
$M$ is a graded free $R$-module if and only if there exists homogeneous elements $e_{\lambda} \in M$ ($\lambda \in \Lambda$), such that $\{e_{\lambda},\lambda \in \Lambda$} is a linearly Independent generating set for $M$.In this case we have
\begin{equation} M = \bigoplus\limits_{\lambda \in \Lambda}R_{\lambda} \cong \bigoplus\limits_{\lambda \in \Lambda}R(-d_{\lambda})~~~~~~~~~\text{(1)} \end{equation} where $R_{\lambda}$ is a copy of $R$ and $d_{\lambda} = \text{deg}~ e_{\lambda}$ .
I don't understand why the relationship (1) holds. Can someone explain it please?