Math 2.0

Commutative algebra Proposition

Prime elements are irreducible in an integral domain

Builds on Prime elementIrreducible elementIntegral domain

Statement

Let RR be an integral domain and let p∈Rp \in R be a prime element. Then pp is irreducible.

Remarks

The proof: let p=abp = a b. Then p∣abp \mid a b, so p∣ap \mid a, say. Write a=pca = p c. Then p=pcbp = p c b, and cancellation gives cb=1c b = 1, so bb is a unit. The converse is false in general: in ℤ[−5]\mathbb{Z}[\sqrt{-5}], 22 is irreducible but not prime. In a unique factorization domain, irreducible elements are prime.

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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 domaindefinitionPrime element, definition, no evidence levelPrime elementdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionPrime elements are irreducible in an integral domain, statement, no evidence levelPrime elements are ir…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.
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  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. Prime element definition
  9. Irreducible element definition
  10. Prime elements are irreducible in an integral domain statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Prime_element, Irreducible elements
About this page
Tag0F0K8DB9TD
URIm2://0F0K8DB9TD
Kindstatement, proposition
WritingW0 stub
MSC13A05
StatusThis copy of the site has no build data. Run m2 status 0F0K8DB9TD.
UsesPrime element, Irreducible element, Integral domain

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