proof that a Euclidean domain is a PID

Let D be a Euclidean domainMathworldPlanetmath, and let 𝔞D be a nonzero ideal. We show that 𝔞 is principal. Let


be the set of Euclidean valuations of the non-zero elements of 𝔞. Since A is a non-empty set of non-negative integers, it has a minimum m. Choose d𝔞 such that ν(d)=m. Claim that 𝔞=(d). Clearly (d)𝔞. To see the reverse inclusion, choose x𝔞. Since D is a Euclidean domain, there exist elements y,rD such that


with ν(r)<ν(d) or r=0. Since r𝔞 and ν(d) is minimal in A, we must have r=0. Thus d|x and x(d).

Title proof that a Euclidean domain is a PID
Canonical name ProofThatAEuclideanDomainIsAPID
Date of creation 2013-03-22 12:43:11
Last modified on 2013-03-22 12:43:11
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 7
Author rm50 (10146)
Entry type Result
Classification msc 13F07
Related topic PID
Related topic UFD
Related topic IntegralDomain
Related topic EuclideanValuation