= Prime ideals of a commutative Artinian ring
A commutative <Artinian ring> has finitely many <prime ideals>, and every one is a <maximal ideal>. If $\mathfrak p$ is prime, the Artinian domain $R/\mathfrak p$ is a field: stabilization of $(x)\supseteq(x^2)\supseteq\cdots$ gives $x^n=x^{n+1}a$, and cancellation gives $xa=1$. Infinitely many distinct maximal ideals $\mathfrak m_i$ would give a strictly descending chain
$$
\mathfrak m_1\supsetneq\mathfrak m_1\mathfrak m_2\supsetneq
\mathfrak m_1\mathfrak m_2\mathfrak m_3\supsetneq\cdots,
$$
because distinct maximal ideals are comaximal.
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