Proposition 1

If every element of $\Delta$ is cancellable and $(A_\lambda)$ is a graduation of type $\Delta$ compatible with the structure of a ring $A$, $A_0$ is a subring of $A$ (and in particular $1 \in A_0$).

As $A_0 A_0 \subset A_0$ by definition, it suffices to prove that $1 \in A_0$. Let $1 = \sum_{\lambda \in \Delta} e_\lambda$ be the decomposition of 1 into its homogeneous components. If $x \in A_\mu$, then $x = x . 1 = \sum_{\lambda \in \Delta} x e_\lambda$; comparing the components of degree $\mu$, (since $\mu + \lambda = \mu$ implies $\lambda = 0$) $x = x e_0$. Since this relation is true for every homogeneous element of $A$, it is true for all $x \in A$; in particular $1 = 1 . e_0 = e_0 \in A_0$.