Math 2.0

statement theorem

PID implies UFD

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.

Explanations

L1 human

In a principal ideal domain, every irreducible element behaves like a prime number. That is the key to unique factorisation. So every Euclidean domain, such as the integers, has unique factorisation.

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To read this node, read these 15 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Unique_factorization_domain, Examples

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