PID implies UFD: Difference between revisions
(New page: {{integral domain property implication| stronger = principal ideal domain| weaker = unique factorization domain}} ==Statement== Any principal ideal domain is a [[unique factorization...) |
|||
| Line 12: | Line 12: | ||
# [[uses::Noetherian implies ACCP]] | # [[uses::Noetherian implies ACCP]] | ||
# [[uses::ACCP implies every element has a factorization into irreducibles]] | # [[uses::ACCP implies every element has a factorization into irreducibles]] | ||
# [[uses:: | # [[uses::Principal ideal implies Bezout]] | ||
# [[uses::Bezout implies every irreducible is prime]] | # [[uses::Bezout implies every irreducible is prime]] | ||
# [[uses::Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates]] | # [[uses::Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates]] | ||
Latest revision as of 16:33, 1 February 2009
This article gives the statement and possibly, proof, of an implication relation between two integral domain properties. That is, it states that every integral domain satisfying the first integral domain property (i.e., principal ideal domain) must also satisfy the second integral domain property (i.e., unique factorization domain)
View all integral domain property implications | View all integral domain property non-implications |Get help on looking up integral domain property implications/non-implications
Get more facts about principal ideal domain|Get more facts about unique factorization domain
Statement
Any principal ideal domain is a unique factorization domain.
Facts used
- Principal ideal implies Noetherian
- Noetherian implies ACCP
- ACCP implies every element has a factorization into irreducibles
- Principal ideal implies Bezout
- Bezout implies every irreducible is prime
- Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates
Proof
Facts (1)-(3) guarantee the existence of a factorization into irreducibles for any nonzero non-unit. Facts (4)-(6) guarantee the uniqueness of factorization.