Noetherian ring
If "R" is an integral domain, then formula_111 is also an integral domain; its field of fractions is the field of rational functions. If "R" is a Noetherian ring, then formula_111 is a Noetherian ring. If "R" is a unique factorization domain, then formula_111 is a unique factorization domain. Finally, "R" is a field if and only if formula_111 is a principal ideal domain.