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What is an example of a commutative ring $R$ (with 1) such that dim $R \geq 2$ and there is a unique prime ideal of a given height.

Such a ring is of course local and if we assume $R$ is finite dimensional then it will have only finitely many prime ideals. Can one say something about local rings with finitely many prime ideals. Assume Noetherian if you have to.

Geometrically this translates to finding an affine scheme with a unique irreducible subscheme for a given codimension.

EDIT : This stack exchange question answers mine , in fact is more general.

iron feliks
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1 Answers1

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It is worth mentioning that in the question you link to, the ring is assumed to be noetherian. On the other hand, in the non-noetherian case, we have the following:

Example. Let $V$ be a valuation ring. Then, the prime ideals of $V$ are totally ordered, i.e., they look like $$0 \subsetneq \mathfrak{p}_1 \subsetneq \mathfrak{p}_2 \subsetneq \cdots \subsetneq \mathfrak{m}$$ by [BouCA, VI.1.2, Thm. 1]. Thus, every valuation ring has a unique prime ideal (if it exists) of a given height.

Takumi Murayama
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