PID and UFD are equivalent in a Dedekind domain

This article shows that if A is a Dedekind domainMathworldPlanetmath, then A is a UFD if and only if it is a PID. Note that this result implies the more specific result given in the article unique factorization and ideals in ring of integers.

Since any PID is a UFD, we need only prove the other direction. So assume A is a UFD, let 𝔭 be a nonzero (proper) prime idealMathworldPlanetmathPlanetmath, and choose 0x𝔭. Note that x is a nonunit since 𝔭 is a proper idealMathworldPlanetmath. Since A is a UFD, we may write x uniquely (up to units) as x=p1a1pkak where the pi are distinct irreduciblesPlanetmathPlanetmath in A, the ai are positive integers, and k>0 since x is not a unit. Since 𝔭 is prime and x𝔭, it follows that some pi, say p1, is in 𝔭. Then (p1)𝔭. But (p1) is prime since clearly in a UFD any ideal generated by an irreducible is prime. Since A is Dedekind and thus has Krull dimension 1, it must be that (p1)=𝔭 and thus 𝔭 is principal.

Title PID and UFD are equivalent in a Dedekind domain
Canonical name PIDAndUFDAreEquivalentInADedekindDomain
Date of creation 2013-03-22 17:53:45
Last modified on 2013-03-22 17:53:45
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 7
Author rm50 (10146)
Entry type Theorem
Classification msc 16D25
Classification msc 13G05
Classification msc 13A15
Classification msc 11N80