Math 2.0

Commutative algebra Local rings Counterexample

Nagata's example of a Noetherian ring of infinite dimension

Builds on Noetherian ringKrull dimensionPolynomial ringLocalizationMultiplicative setPrime idealHeight of a prime ideal

Counterexample

Let kk be a field and let A=k[x1,x2,x3,…]A = k[x_1, x_2, x_3, \ldots] be the polynomial ring in countably many variables. Split the variables into consecutive blocks B1,B2,B3,…B_1, B_2, B_3, \ldots where the block BiB_i has exactly ii variables. Let 𝔭i\mathfrak{p}_i be the prime ideal of AA generated by the variables of BiB_i. Let S=Aβˆ–β‹ƒiβ‰₯1𝔭iS = A \setminus \bigcup_{i \ge 1} \mathfrak{p}_i, a multiplicative set, and let R=Sβˆ’1AR = S^{-1} A be the localization. Then RR is a Noetherian ring, but its Krull dimension is infinite.

Remarks

The maximal ideals of RR are exactly the ideals 𝔭iR\mathfrak{p}_i R. The localization of RR at 𝔭iR\mathfrak{p}_i R is a localization of a polynomial ring in ii variables over the field generated by the other variables. So it is Noetherian of dimension ii. Each nonzero element of RR involves finitely many variables, so it lies in only finitely many maximal ideals. A ring whose localizations at maximal ideals are all Noetherian and whose nonzero elements lie in finitely many maximal ideals is Noetherian. The height of 𝔭iR\mathfrak{p}_i R is ii, so the dimension of RR is at least ii for every ii. So the statement "a Noetherian ring has finite Krull dimension" is false, although every Noetherian local ring has finite dimension.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Height of a prime ideal, concept, no evidence levelHeight of a prime ide…conceptKrull dimension, concept, no evidence levelKrull dimensionconceptLocalization, concept, no evidence levelLocalizationconceptMultiplicative set, concept, no evidence levelMultiplicative setconceptNoetherian ring, concept, no evidence levelNoetherian ringconceptPolynomial ring, concept, no evidence levelPolynomial ringconceptPrime ideal, concept, no evidence levelPrime idealconceptNagata's example of a Noetherian ring of infinite dimension, counterexample, no evidence levelNagata's example of a…counterexample
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.
  • no evidence level
  1. Height of a prime ideal concept
  2. Krull dimension concept
  3. Localization concept
  4. Multiplicative set concept
  5. Noetherian ring concept
  6. Polynomial ring concept
  7. Prime ideal concept
  8. Nagata's example of a Noetherian ring of infinite dimension counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Krull_dimension
About this page
TagYEJVKCY78E
URIm2://YEJVKCY78E
Kindcounterexample
WritingW0 stub
MSC13C15, 13E05
UsesNoetherian ring, Krull dimension, Polynomial ring, Localization, Multiplicative set, Prime ideal, Height of a prime ideal
Changed2 November 2026

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