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.
Furthermore, a commutative fir is precisely a principal ideal domain, while a commutative semifir is precisely a Bézout domain.
These last facts are not generally true for noncommutative rings, however (Cohn 1971).
Another important and motivating example of a free ideal ring are the free associative (unital) k-algebras for division rings k, also called non-commutative polynomial rings (Cohn 2000, §5.4).