Lemma 15.42.2. Let $\varphi : R \to S$ be a ring map. Assume

$\varphi $ is regular,

$S$ is Noetherian, and

$R$ is Noetherian and normal.

Then $S$ is normal.

Lemma 15.42.2. Let $\varphi : R \to S$ be a ring map. Assume

$\varphi $ is regular,

$S$ is Noetherian, and

$R$ is Noetherian and normal.

Then $S$ is normal.

**Proof.**
For Noetherian rings being normal is the same as having properties $(S_2)$ and $(R_1)$, see Algebra, Lemma 10.157.4. Hence we may apply Algebra, Lemmas 10.163.4 and 10.163.5.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #4259 by DS on

Comment #4429 by Johan on