Principal ideal domain

In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element).

Examples are the ring of integers, all fields, and rings of polynomials in one variable with coefficients in a field. All euclidean domains are principal ideal domains, but the converse is not true. The ring Z[X] of all polynomials with integer coefficients is not principal, since for example the ideal generated by 2 and X cannot be generated by a single polynomial.

In a principal ideal domain, any two elements have a greatest common divisor (and may have more than one).

In all rings, maximal ideals are prime. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.

Every principal ideal domain is Noetherian and a unique factorization domain. The ring K[X,Y] for any field K is a UFD but is not a PID.

An example of a principal ideal domain that is not a euclidean domain is the ring (Wilson, J. C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag. 34-38, 1973).