Corollary

If every element of $\Delta$ is cancellable, the centralizer of the graded subring $B$ of $A$ (and in particular the centre of $A$) is a graded subring of $A$.

It is the intersection of the centralizers of the homogeneous elements of $B$.

Remark (3) *A direct system $(A_\alpha, \phi_{\beta\alpha})$ of graded rings of type $\Delta$ (resp. a direct system $(M_\alpha, f_{\beta\alpha})$ of graded $A_\alpha$-modules of type $\Delta$) is a direct system of rings (resp. $A_\alpha$-modules) such that each $A_\alpha$ (resp. $M_\alpha$) is graded of type $\Delta$ and each $\phi_{\beta\alpha}$ (resp. $f_{\beta\alpha}$) is a homomorphism of graded rings (resp. an $A_\alpha$-homomorphism of degree 0 of graded modules). If $(A^\lambda_\alpha)_{\lambda \in \Delta}$ (resp. $(M^\lambda_\alpha)_{\lambda \in \Delta}$) be the graduation of $A_\alpha$ (resp. $M_\alpha$) and we write

it follows from § 6, no. 2, Proposition 5 that $(A^\lambda)$ (resp. $(M^\lambda)$) is a graduation of A (resp. M) and it follows from I, § 10, nos. 3 and 4 that this graduation is compatible with the ring structure on A (resp. the A-module structure on M). The graded ring A (resp. graded A-module M) is called the direct limit of the direct system of graded rings $(A_\alpha, \phi_{\beta\alpha})$ (resp. graded modules $(M_\alpha, f_{\beta\alpha})$). If $\phi_\alpha : A_\alpha \to A$ (resp. $f_\alpha : M_\alpha \to M$) is the canonical mapping, $\phi_\alpha$ (resp. $f_\alpha$) is a homomorphism of graded rings (resp. a homomorphism of degree 0 of graded $A_\alpha$-modules).