Proposition 14

Let $A$ be a noetherian ring. The following conditions are equivalent:

(i) $A$ is reduced and integrally closed in its total ring of fractions ; (ii) $A$ is isomorphic to the product of a finite family of integrally closed rings ; (iii) the canonical components of $A$ are integrally closed ; (iv) for every prime ideal (resp. maximal ideal) $p$ of $A$, the ring $A_p$ is integrally closed.

The equivalence of (i) and (ii) follows from V, § 1, No. 2, Cor. 2 of Prop. 9, and that of (ii) and (iii) from the remarks preceding the proposition. Let $p$ be a prime ideal of $A$; there exists a unique canonical component $A'$ of $A$ such that $p$ belongs to the closed subset $\mathrm{Spec}(A')$ of $\mathrm{Spec}(A)$, and one has a canonical isomorphism $A_p \to A'_pA'$. The equivalence of (iii) and (iv) therefore follows from loc. cit., No. 5, Cor. 1 and Cor. 3 of Prop. 16.

A ring $A$ is said to be normal if it is noetherian and satisfies the equivalent conditions (i) to (iv) of Proposition 14. A noetherian ring is integrally closed if and only if it is integral and normal. A normal local ring is integrally closed.