(abbrev. pid) or principal domain, n. an integral domain all of whose ideals are principal ideals; then prime and irreducible elements coincide, and the domain is a unique factorization domain. There are exactly nine imaginary quadratic fields Q(√d) whose subring of algebraic integers yield principal ideal domains, those with -d=1, 2, 3, 7, 11, 19, 43, 67, or 163.