Math 2.0

Commutative algebra Theorem

In a UFD irreducible elements are prime

Builds on Unique factorisation domainIrreducible elementPrime elementPrime elements are irreducible in an integral domain

Statement

Let RR be a unique factorisation domain and let p∈Rp \in R. Then pp is irreducible if and only if pp is prime.

Remarks

Prime elements are irreducible in every integral domain. The other direction needs unique factorisation. The proof: suppose pp is irreducible and pc=abp c = a b. Factor aa, bb and cc into irreducibles. By uniqueness, pp is an associate of an irreducible factor of aa or of bb. Conversely, an integral domain in which every nonzero nonunit is a product of irreducibles, and every irreducible is prime, is a unique factorisation domain.

Reading path

To understand this page from the ground up, read these 13 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 ringdefinitionUnit 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…definitionIntegral domain, definition, no evidence levelIntegral domaindefinitionAssociate elements, definition, no evidence levelAssociate elementsdefinitionPrime element, definition, no evidence levelPrime elementdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionUnique factorisation domain, definition, no evidence levelUnique factorisation …definitionPrime elements are irreducible in an integral domain, statement, no evidence levelPrime elements are ir…statementIn a UFD irreducible elements are prime, statement, no evidence levelIn a UFD irreducible …statement
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. Unit of a ring definition
  5. Zero divisor definition
  6. Divisibility in a commutative ring definition
  7. Integral domain definition
  8. Associate elements definition
  9. Prime element definition
  10. Irreducible element definition
  11. Unique factorisation domain definition
  12. Prime elements are irreducible in an integral domain statement
  13. In a UFD irreducible elements are prime statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Unique_factorization_domain, Properties
About this page
Tag0QRCFZKXH8
URIm2://0QRCFZKXH8
Kindstatement, theorem
WritingW0 stub
MSC13F15
StatusThis copy of the site has no build data. Run m2 status 0QRCFZKXH8.
UsesUnique factorisation domain, Irreducible element, Prime element, Prime elements are irreducible in an integral domain

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