Math 2.0

Commutative algebra Definition

Krull domain

Builds on Integral domainField of fractionsDiscrete valuation ring

Definition

Let AA be an integral domain with field of fractions KK. A discrete valuation of KK is a surjective map v ⁣:K×→ℤv \colon K^\times \to \Z with v(xy)=v(x)+v(y)v(xy) = v(x) + v(y) and v(x+y)≥min⁡(v(x),v(y))v(x + y) \ge \min(v(x), v(y)) whenever x,y,x+y∈K×x, y, x + y \in K^\times; its ring is Rv={0}∪{x∈K×:v(x)≥0}R_v = \set{0} \cup \set{x \in K^\times : v(x) \ge 0}, a discrete valuation ring. AA is a Krull domain if there is a family (vi)i∈I(v_i)_{i \in I} of discrete valuations of KK such that:

  1. A=⋂i∈IRviA = \bigcap_{i \in I} R_{v_i};
  2. for every x∈K×x \in K^\times, vi(x)≠0v_i(x) \ne 0 for only finitely many i∈Ii \in I.

Remarks

Example: ℤ\Z is a Krull domain, with the pp-adic valuations vpv_p of ℚ\Q for all primes pp. A rational number xx lies in ℤ\Z exactly when vp(x)≥0v_p(x) \ge 0 for all pp, and a nonzero rational number involves only finitely many primes. More generally every unique factorisation domain and every Noetherian integrally closed domain is a Krull domain. Non-example: ℤ[−3]\Z[\sqrt{-3}] is not a Krull domain. An intersection of valuation rings is integrally closed, but (1+−3)/2(1 + \sqrt{-3})/2 is a root of x2−x+1x^2 - x + 1 and does not lie in ℤ[−3]\Z[\sqrt{-3}]. So clause 1 fails for every family. The family can be chosen as the valuations of the localizations APA_P at the height-one primes PP of AA. By the Mori–Nagata theorem, the integral closure of a Noetherian domain is a Krull domain.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Discrete valuation ring, concept, no evidence levelDiscrete valuation ri…conceptGroup, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionField of fractions, definition, no evidence levelField of fractionsdefinitionKrull domain, definition, no evidence levelKrull domaindefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Discrete valuation ring concept
  2. Group concept
  3. Ring definition
  4. Commutative ring definition
  5. Unit of a ring definition
  6. Zero divisor definition
  7. Field definition
  8. Integral domain definition
  9. Field of fractions definition
  10. Krull domain definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Krull_ring
About this page
Tag8KK1BQSPJT
URIm2://8KK1BQSPJT
Kinddefinition
WritingW0 stub
MSC13F05
DefinesKrull domain
UsesIntegral domain, Field of fractions, Discrete valuation ring
Changed2 November 2026

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