Remark 1

A convenient form of th. 4 is the following: let $A$ be a noetherian ring, such that for every prime ideal $p$ of $A$ the ring $A_p$ is integrally closed or of depth $\geqslant 2$; then $A$ is normal. Let us indeed verify the assumptions of th. 4. Let $p \in \mathrm{Spec}(A)$. If $\mathrm{ht}(p) \leqslant 1$, then one has $\mathrm{prof}(A_p) \leqslant 1$, hence $A_p$ is integrally closed. If $\mathrm{ht}(p) \geqslant 2$, the ring $A_p$ is either of depth $\geqslant 2$, or integrally closed, which again implies $\mathrm{prof}(A_p) \geqslant 2$ (No. 1, cor. 2 of prop. 1), whence the desired conclusion. One verifies analogously the following statement: let $A$ be a noetherian ring such that for every prime ideal $p$ of $A$, the ring $A_p$ is reduced or of depth $\geqslant 1$; then $A$ is reduced.