The Stacks project

Proof. Assume $R$ is a Noetherian ring of dimension $0$. By Lemma 10.31.5 the space $\mathop{\mathrm{Spec}}(R)$ is Noetherian. By Topology, Lemma 5.9.2 we see that $\mathop{\mathrm{Spec}}(R)$ has finitely many irreducible components, say $\mathop{\mathrm{Spec}}(R) = Z_1 \cup \ldots \cup Z_ r$. According to Lemma 10.26.1 each $Z_ i = V(\mathfrak p_ i)$ with $\mathfrak p_ i$ a minimal ideal. Since the dimension is $0$ these $\mathfrak p_ i$ are also maximal. Thus $\mathop{\mathrm{Spec}}(R)$ is the discrete topological space with elements $\mathfrak p_ i$. All elements $f$ of the Jacobson radical $\bigcap \mathfrak p_ i$ are nilpotent since otherwise $R_ f$ would not be the zero ring and we would have another prime. By Lemma 10.53.5 $R$ is equal to $\prod R_{\mathfrak p_ i}$. Since $R_{\mathfrak p_ i}$ is also Noetherian and dimension $0$, the previous arguments show that its radical $\mathfrak p_ iR_{\mathfrak p_ i}$ is locally nilpotent. Lemma 10.32.5 gives $\mathfrak p_ i^ nR_{\mathfrak p_ i} = 0$ for some $n \geq 1$. By Lemma 10.52.8 we conclude that $R_{\mathfrak p_ i}$ has finite length over $R$. Hence we conclude that $R$ is Artinian by Lemma 10.53.6.

If $R$ is an Artinian ring then by Lemma 10.53.6 it is Noetherian. All of its primes are maximal by a combination of Lemmas 10.53.3, 10.53.4 and 10.53.5. $\square$