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.
... Furthermore, a commutative fir is precisely a principal ideal domain, while a commutative semifir is precisely a Bézout domain ... Every principal right ideal domain R is a right fir, since every nonzero principal right ideal of a domain is isomorphic to R ... Since all right ideals of a right fir are free, they are projective ...
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