Exercise 3, § 4

A ring $A$ is said to be normal if the ring $A_p$ is integrally closed for every prime ideal $p$ of $A$. a) A normal ring $A$ is reduced, and every prime ideal $p$ of $A$ contains only one minimal prime ideal of $A$. For every minimal prime ideal $p$ of $A$, the ring $A/p$ is integrally closed. b) A noetherian normal ring $A$ is isomorphic to a product of a finite number of integrally closed noetherian rings. c) Let $A$ be a noetherian normal ring and $a$ an element of $A$. Then $\operatorname{Ass}_A(A/aA)$ contains no embedded prime ideal (IV, § 2, no 3, remark).

EXERCISES

AC IX.79