# Free ideal ring

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.

## Properties and examples

It turns out that a left and right fir is a domain. 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).

Every principal right ideal domain *R* is a right fir, since every nonzero principal right ideal of a domain is isomorphic to *R*. In the same way, a right Bézout domain is a semifir.

Since all right ideals of a right fir are free, they are projective. So, any right fir is a right hereditary ring, and likewise a right semifir is a right semihereditary ring. Because projective modules over local rings are free, and because local rings have invariant basis number, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a right semifir.

Unlike a principal right idea domain, a right fir is not necessarily right Noetherian, however in the commutative case, *R* is a Dedekind domain since it is a hereditary domain, and so is necessarily Noetherian.

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).

Semifirs have invariant basis number and every semifir is a Sylvester domain.

## References

- Cohn, P. M. (1971), "Free ideal rings and free products of rings",
*Actes du Congrès International des Mathématiciens (Nice, 1970)*,**1**, Gauthier-Villars, pp. 273–278, MR 0506389 - Cohn, P. M. (2006),
*Free ideal rings and localization in general rings*, New Mathematical Monographs,**3**, Cambridge University Press, ISBN 978-0-521-85337-8, MR 2246388 - Cohn, P. M. (1985),
*Free rings and their relations*, London Mathematical Society Monographs,**19**(2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-179152-0, MR 0800091 - Cohn, P. M. (2000),
*Introduction to ring theory*, Springer Undergraduate Mathematics Series, Berlin, New York: Springer-Verlag, ISBN 978-1-85233-206-8, MR 1732101 - Hazewinkel, Michiel, ed. (2001) [1994], "F/f041580",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

## Further reading

- Cohn, P.M. (1995),
*Skew fields. Theory of general division rings*, Encyclopedia of Mathematics and Its Applications,**57**, Cambridge: Cambridge University Press, ISBN 0-521-43217-0, Zbl 0840.16001