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).
$\square$

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## Comments (2)

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