Corollary 3

Let $A = \bigoplus_{i \in \mathbf{Z}} A_i$ be an integrally closed graded domain. Then, for every integer $d > 0$, the ring $A^{(d)}$ (Chapter III, § 1, no. 3) is an integrally closed domain.

Let $U$ be the set of homogeneous elements $\neq 0$ of $A^{(d)}$ and let $x$ be a homogeneous element of $U^{-1}A^{(d)}$ integral over $A^{(d)}$ and hence over $A$; as $x \in S^{-1}A$, $x$ belongs to $A$ by hypothesis; as its degree is divisible by $d$, it belongs to $A^{(d)}$ and it then follows from Corollary 1 that $A^{(d)}$ is integrally closed.