# PID

A principal ideal domain is an integral domain where every ideal is a principal ideal.

In a PID, an ideal $(p)$ is maximal if and only if $p$ is irreducible (and prime since any PID is also a UFD (http://planetmath.org/PIDsAreUFDs)).

Note that subrings of PIDs are not necessarily PIDs. (There is an example of this within the entry biquadratic field.)

 Title PID Canonical name PID Date of creation 2013-03-22 11:56:25 Last modified on 2013-03-22 11:56:25 Owner mps (409) Last modified by mps (409) Numerical id 13 Author mps (409) Entry type Definition Classification msc 16D25 Classification msc 13G05 Classification msc 11N80 Classification msc 13A15 Synonym principal ideal domain Related topic UFD Related topic Irreducible Related topic Ideal Related topic IntegralDomain Related topic EuclideanRing Related topic EuclideanValuation Related topic ProofThatAnEuclideanDomainIsAPID Related topic WhyEuclideanDomains