Exercise 24, § 4

Let $A$ be a semilocal noetherian ring, $\hat{A}$ its completion, $x$ a nonzero element of the radical of $A$. Suppose that $\mathrm{Ass}_A(A/xA)$ contains no embedded prime ideal and that, for every $p \in \mathrm{Ass}_A(A/xA)$, the ring $A_p$ is regular and the ring $\kappa(p) \otimes_A \hat{A}$ reduced. Then $\hat{A}$ is reduced. (Reason as in the proof of Theorem 3, (i) $\Rightarrow$ (ii) of § 4, No. 4.)

EXERCISES