I am reading graded modules. I understood the definition, but I am not getting the following point.
Let $R$ be a graded ring of type $\triangle$, $M$ a graded $R$-module of type $\triangle$, where $\triangle$ is a commutative monoid. For $\lambda_0 \in \triangle$, let $M'(\lambda)=M_{\lambda + \lambda_0}$, then $M'=\oplus_{\lambda \in \triangle} M' _{\lambda}$ is again a graded $R$ module. I understood this much, but at the end there is a line," When $\triangle$ is a group, the underlying $R$-module of the graded $R$-module $M'$ is identified with $M$". This is not clear to me, can someone explain it, please?
I believe the following is meant: When $\Delta$ is a group, then in particular, we know that $\lambda_0$ is invertible, so that there exists some $-\lambda_0 \in \Delta$ such that $\lambda_0 + (-\lambda_0) = 0$. From this we obtain an isomorphism $$ M \to M', M_\lambda \ni x \mapsto x \in M_{\lambda_0 + (\lambda - \lambda_0)}. $$
Example: Suppose $\Delta = \mathbb N_0$. This is not a group. Set $\lambda_0 = 1$. Suppose $M_0 = \mathbb Z$ and $M_k = 0$ for $k \ge 1$. Then $M' \cong 0$, whereas $M \cong \mathbb Z$.