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). $\square$

Comment #3421 by Jonas Ehrhard on

We should assume that the $\mathfrak{m}_i$ are pairwise different to apply the Chinese Remainder theorem.

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