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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.

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  • $\begingroup$ I'm afraid there is no such characterisation. A discrete valuation ring has dimension one, and there's a characterisation of Dedekind domains in terms of dimension (among other properties). $\endgroup$ – Bernard Jun 16 '16 at 22:20
  • $\begingroup$ This is way too broad. Maybe make some further assumptions like noetherian domains of Krull dimension 1. $\endgroup$ – MooS Jun 17 '16 at 9:03
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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.

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