Corollary 1

Every regular noetherian local ring is integrally closed, and in particular an integral domain.

Suppose that A is regular. Then $\mathrm{gr}(A)$ is isomorphic to a polynomial algebra in a finite number of indeterminates over a field (Theorem 1, (iv)). Hence $\mathrm{gr}(A)$ is a noetherian integrally closed ring (V, § 1, No. 3, Corollary 3 to Proposition 13), and therefore A is integrally closed (V, § 1, No. 4, Proposition 15).

We shall see in a later chapter that every regular noetherian local ring is factorial.