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Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can this fail if $R$ is a non-Noetherian ring? For example, if $R$ is non-Noetherian, is it possible for there to be a minimal prime over $(r)$ of infinite height?

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2 Answers 2

up vote 13 down vote accepted

I think that the answer is yes.

Indeed, there are examples of integral domains $D$ such that every non-zero prime ideal of $D$ has infinite height.

Look at the paper

"Anti-archimedean rings and power series rings"

D.D. Anderson; B.G. Kang; M H. Park

Communications in Algebra, 1532-4125, Volume 26, Issue 10, 1998, Pages 3223 – 3238.

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Thanks! Amazingly, your answer also takes care of my motivation for this question. I had been wondering how a ring might have every prime be of the same height, and it's clear that the only options are 0 and $\infty$; but since there must always be minimal primes, $\infty$ is technically impossible, but we can (WLOG) let 0 be a prime of height 0 and see if we can get the rest to be the same. So I wanted to check that even primes we'd normally expect to be small (e.g. minimal primes over a principal ideal) could still be of infinite height. –  Zev Chonoles Oct 17 '10 at 16:26

Valuation rings demonstrate quite clearly the failure of Krull's principal ideal theorem: take a valuation ring O of finite dimension. The prime ideals then form a chain

$p_0:=0\subset p_1\subset\ldots\subset p_d$

so that for every $i\in\{1,\ldots ,d\}$ there exists $r_i\in p_i\setminus p_{i-1}$. Obviously $p_i$ is a minimal prime over $r_iO$.

For valuation domains of infinite dimension one has to consider the so-called limit-primes: a prime ideal $p$ of a commutative ring $R$ is called limit-prime if

$p=\bigcup\limits_{q\in\mathrm{Spec} (R): q\subset p}q$.

There exist valuation domains $O$ of infinite Krull dimension such that the maximal ideal $m$ of $O$ is no limit-prime. For example take a valuation ring such that the corresponding value group is

$\mathbb{Z}\times\mathbb{Z}\times\ldots$ (countably many factors ordered lexigraphically).

Then one can find $r\in m$ such that $m$ is minimal over $rO$.

H

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+1 Thanks for your enlightening examples. What would be the best place to learn more about not-necessarily-Noetherian valuation rings? –  Zev Chonoles Oct 20 '10 at 1:40
    
Valuation domains are frequently studied via the associated Krull valuations, so that any book on valuation theory can be used. Personally I learned a lot from Otto Endler's Valuation Theory and Zariski-Samuel, Commutative Algebra Vol. 2. Moreover Robert Gilmer's book on Multiplicative Ideal Theory and Fuchs-Salce, Modules over valuation domains. H –  Hagen Oct 20 '10 at 8:08

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