The Stacks project

Lemma 10.61.2. A Noetherian ring with finitely many primes has dimension $\leq 1$.

Proof. Let $R$ be a Noetherian ring with finitely many primes. If $R$ is a local domain, then the lemma follows from Lemma 10.61.1. If $R$ is a domain, then $R_\mathfrak m$ has dimension $\leq 1$ for all maximal ideals $\mathfrak m$ by the local case. Hence $\dim (R) \leq 1$ by Lemma 10.60.4. If $R$ is general, then $\dim (R/\mathfrak q) \leq 1$ for every minimal prime $\mathfrak q$ of $R$. Since every prime contains a minimal prime (Lemma 10.17.2), this implies $\dim (R) \leq 1$. $\square$

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