Math 2.0

Commutative algebra Definition

Unique factorisation domain

Builds on Integral domainIrreducible elementAssociate elements

Definition

A unique factorisation domain is an integral domain RR with these two properties.

  1. Every nonzero element a∈Ra \in R that is not a unit is a product a=p1p2⋯pna = p_1 p_2 \cdots p_n of irreducible elements, with n≥1n \ge 1.
  2. The factorization is unique up to order and units. If p1⋯pn=q1⋯qmp_1 \cdots p_n = q_1 \cdots q_m with all pi,qjp_i, q_j irreducible, then n=mn = m. Also, there is a permutation σ\sigma of {1,…,n}\{1, \dots, n\} such that pip_i and qσ(i)q_{\sigma(i)} are associates for each ii.

Remarks

The integers are a unique factorisation domain: this is the fundamental theorem of arithmetic. Every principal ideal domain is a unique factorisation domain, and so is every polynomial ring over a unique factorisation domain. The ring ℤ[−5]\mathbb{Z}[\sqrt{-5}] is not a unique factorisation domain. In a unique factorisation 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 domaindefinitionAssociate elements, definition, no evidence levelAssociate elementsdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionUnique factorisation domain, definition, no evidence levelUnique factorisation …definition
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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. Associate elements definition
  9. Irreducible element definition
  10. Unique factorisation domain definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Unique_factorization_domain, Definition
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DefinesUnique factorisation domain
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