Rings of Krull dimension one

Question

I have to write a monograph about commutative rings with Krull dimension $1$, but I can't find results, so I am looking foward for some references, and some results to search. Also, I would appreciate a lot to know if there is some result of the kind:

$$ \dim(A)=1 \iff ~?$$

Thanks in advance.

Answer 1

While there is no characterization of one-dimensional rings, there are various theorems involving them. Here is a small sample, off the top of my head.

1. A UFD is a PID if and only if it is one-dimensional.
2. A valuation domain is completely integrally closed if and only if it is a one-dimensional. Therefore a one-dimensional Prufer domain is completely integrally closed. (But the converse is false. For example, the ring of integer-valued polynomials is a 2-dimensional completely integrally closed Prufer domain.)
3. One-dimensional domains have almost stable rank 1.
4. One-dimensional Bezout domains are elementary divisor rings.

Edited: Accidentally wrote "rings" instead of "domains" in a couple spots. Corrected an error pointed out in the comments.

Answer 2

An integral extension $S$ of a ring $R$ of Krull dimension $1$ has Krull dimension $1$. This is because any 3-chain of prime ideals $P_1 \subsetneq P_2 \subsetneq P_3$ induces an inclusion $p_1 \subset p_2 \subset p_3$ (where we define $p_j := R \cap P_j$ for $j = 1, 2, 3$), of which not all inclusions can be proper because of what we assumed about $R$, and then we only have to use that between two prime ideals lying over the same prime ideal there are no proper inclusions.

This, of course, generalises to show that the Krull dimension is constant when passing to an integral extension.