Proposition 5

Let $A$ be a graded ring of type $\Delta$. If every element of $\Delta$ is cancellable, then, for every homogeneous element $a \in A$, the centralizer of $a$ in $A$ (I, § 1, no. 5) is a graded subring of $A$.

Suppose that $a$ is of degree $\delta$; let $b = \sum_\lambda b_\lambda$ be an element permutable with $a$, $b_\lambda$ being the homogeneous component of $b$ of degree $\lambda$ for all $\lambda \in \Delta$. Then by hypothesis $\sum_\lambda (ab_\lambda - b_\lambda a) = 0$ and $ab_\lambda - b_\lambda a$ is homogeneous of degree $\lambda + \delta$; as $\delta$ is cancellable, it follows that $ab_\lambda = b_\lambda a$ for all $\lambda$, which proves our assertion.