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.