The Stacks project

Pure ideals are determined by their vanishing locus.

Lemma 10.108.3. Let $R$ be a ring. If $I, J \subset R$ are pure ideals, then $V(I) = V(J)$ implies $I = J$.

Proof. For example, by property (7) of Lemma 10.108.2 we see that $I = \mathop{\mathrm{Ker}}(R \to \prod _{\mathfrak p \in V(I)} R_{\mathfrak p})$ can be recovered from the closed subset associated to it. $\square$

Comments (1)

Comment #1221 by David Corwin on

Suggested slogan: Pure ideals are determined by their vanishing locus

There are also:

  • 2 comment(s) on Section 10.108: Pure ideals

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04PT. Beware of the difference between the letter 'O' and the digit '0'.