Lemma 15.17.2. Let $R$ be an Artinian ring. Let $M$ be an $R$-module. Let $I \subset R$ be the smallest ideal $I \subset R$ such that $M/IM$ is flat over $R/I$. Then $I$ has the following universal property: For every ring map $\varphi : R \to R'$ we have

**Proof.**
Note that $I$ exists by Lemma 15.17.1. The implication $\Rightarrow $ follows from Algebra, Lemma 10.38.7. Let $\varphi : R \to R'$ be such that $M \otimes _ R R'$ is flat over $R'$. Let $J = \mathop{\mathrm{Ker}}(\varphi )$. By Algebra, Lemma 10.100.7 and as $R' \otimes _ R M = R' \otimes _{R/J} M/JM$ is flat over $R'$ we conclude that $M/JM$ is flat over $R/J$. Hence $I \subset J$ as desired.
$\square$

## Post a comment

Your email address will not be published.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 (0)