Math 2.0

Commutative algebra Definition

Mori domain

Builds on Integral domainField of fractionsIdeal

Definition

Let AA be an integral domain with field of fractions KK. For a nonzero ideal II of AA, let I−1={x∈K:xI⊆A}I^{-1} = \set{x \in K : xI \subseteq A} and Iv={x∈K:xI−1⊆A}I_v = \set{x \in K : x I^{-1} \subseteq A}. The ideal II is divisorial if Iv=II_v = I. AA is a Mori domain if every strictly ascending chain I1⊊I2⊊⋯I_1 \subsetneq I_2 \subsetneq \cdots of nonzero divisorial ideals of AA is finite.

Remarks

Every nonzero principal ideal aAaA is divisorial, since (aA)−1=a−1A(aA)^{-1} = a^{-1}A and then (aA)v=aA(aA)_v = aA. Example: every Noetherian domain is a Mori domain, since it has no infinite strictly ascending chain of ideals at all. Non-example: the ring ℤ‾\overline{\Z} of all algebraic integers is not Mori; the principal ideals (21/2n)(2^{1/2^n}) for n≥1n \ge 1 form a strictly ascending chain, since 21/2n+1/21/2n=2−1/2n+12^{1/2^{n+1}} / 2^{1/2^{n}} = 2^{-1/2^{n+1}} is not an algebraic integer. Every Krull domain is a Mori domain, and a Mori domain is Krull if and only if it is completely integrally closed.

Reading path

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

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionIdeal, definition, no evidence levelIdealdefinitionUnit 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 fractionsdefinitionMori domain, definition, no evidence levelMori 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.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Ideal definition
  5. Unit of a ring definition
  6. Zero divisor definition
  7. Field definition
  8. Integral domain definition
  9. Field of fractions definition
  10. Mori domain definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mori_domain
About this page
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DefinesMori domain
UsesIntegral domain, Field of fractions, Ideal
Changed2 November 2026

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