A principal ideal domain is an integral domain in which every proper ideal can be generated by a single element. The term "principal ideal domain" is often abbreviated P.I.D. Examples of P.I.D.s include the integers, the Gaussian integers, and the set of polynomials in one variable with real coefficients.
Every Euclidean ring is a principal ideal domain, but the converse is not true. Nevertheless, the notion of greatest
common divisor arising from the Euclidean
algorithm can be extended to the more general context of principal ideal domains
as follows. Given two nonzero elements 
of a principal ideal domain 
, a greatest common divisor
of 
and 
is defined as any element 
of 
such that
 |
Every principal ideal domain is a unique factorization domain, but not conversely. Every polynomial ring over a field is a unique factorization domain, but it is a principal ideal domain iff the number of indeterminates is one.
