In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules of unique rank. (Thus, a ring is semifir if it is n-fir for all n = 0.) The semifir property is left-right symmetric, but the fir property is not.