Math 2.0

Commutative algebra Domains and factorization Theorem

PID implies UFD

Builds on Principal ideal domainUnique factorisation domain

Statement

Let RR be a principal ideal domain. Then RR is a unique factorisation domain.

Remarks

Existence of factorisations: a principal ideal domain is Noetherian, so a process of splitting an element into factors must stop. Uniqueness: in a principal ideal domain, an irreducible element generates a maximal ideal, so it is prime. Consequences: ℤ\mathbb{Z}, ℤ[i]\mathbb{Z}[i] and K[x]K[x] for a field KK have unique factorisation. The converse is false: ℤ[x]\mathbb{Z}[x] is a unique factorisation domain but not a principal ideal domain.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

Show the path as a graph
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 divisordefinitionDivisibility in a commutative ring, definition, no evidence levelDivisibility in a com…definitionIdeal generated by a subset, definition, no evidence levelIdeal generated by a …definitionIntegral domain, definition, no evidence levelIntegral domaindefinitionAssociate elements, definition, no evidence levelAssociate elementsdefinitionPrincipal ideal, definition, no evidence levelPrincipal idealdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionUnique factorisation domain, definition, no evidence levelUnique factorisation …definitionPrincipal ideal domain, definition, no evidence levelPrincipal ideal domaindefinitionPID implies UFD, statement, no evidence levelPID implies UFDstatement
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. Divisibility in a commutative ring definition
  8. Ideal generated by a subset definition
  9. Integral domain definition
  10. Associate elements definition
  11. Principal ideal definition
  12. Irreducible element definition
  13. Unique factorisation domain definition
  14. Principal ideal domain definition
  15. PID implies UFD statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Unique_factorization_domain, Examples
About this page
Tag055HT85T3S
URIm2://055HT85T3S
Kindstatement, theorem
WritingW0 stub
MSC13F15
StatusThis copy of the site has no build data. Run m2 status 055HT85T3S.
UsesPrincipal ideal domain, Unique factorisation domain

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