Lemma 10.53.3. If $R$ is Artinian then $R$ has only finitely many maximal ideals.

**Proof.**
Suppose that $\mathfrak m_ i$, $i = 1, 2, 3, \ldots $ are pairwise distinct maximal ideals. Then $\mathfrak m_1 \supset \mathfrak m_1\cap \mathfrak m_2 \supset \mathfrak m_1 \cap \mathfrak m_2 \cap \mathfrak m_3 \supset \ldots $ is an infinite descending sequence (because by the Chinese remainder theorem all the maps $R \to \oplus _{i = 1}^ n R/\mathfrak m_ i$ are surjective).
$\square$

## 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Your email address will not be published.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 (2)

Comment #3421 by Jonas Ehrhard on

Comment #3483 by Johan on

There are also: