existence of infinitely many prime ideals in positive dimension

Question

Let $R$ be a Noetherian ring of Krull dimension $\ge 1$. Is the spectrum of $R$ always a set of infinite cardinality? I know this is the case when $R$ is an affine domain over a field of characteristic zero. What is the most general case that we can establish this statement? Proof or reference (purely algebraic please)?

PS/Edit: The focus of my question is in the most general case where the statement is true.

The ring $\Bbb{Z}_p$ of $p$-adic integers has dimension $1$, is Noetherian and has only 2 prime ideals. In general, any DVR has these properties. — Crostul, Mar 25 '17 at 17:56
Maybe you want $\dim R\ge 2$. In this case your claim holds. — user26857, Mar 25 '17 at 18:42
http://math.stackexchange.com/questions/334373/noetherian-rings-and-prime-ideals; reference Kaplansky, Theorem 144. — user26857, Mar 25 '17 at 18:47
@user26857: Actually your MSE reference satisfies me fully. Nicely explained!! — Manos, Mar 25 '17 at 18:58

score 1 · Answer 1 · answered Mar 25 '17 at 17:59

1

It is false. A discrete valuation domain (viz. $\mathbf Z_{(p)}$ or $K[[X]]$, $K$ a field) has Krull dimension $1$, and its spectrum has two elements.

answered Mar 25 '17 at 17:59

Bernard

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existence of infinitely many prime ideals in positive dimension

1 Answers1