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 {{harv|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 ideal 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 {{harv|Cohn|2000|loc=§5.4}}.
Semifirs have invariant basis number and every semifir is a Sylvester domain.
References
- {{Citation | last1=Cohn | first1=P. M. | authorlink = Paul Cohn| title=Actes du Congrès International des Mathématiciens (Nice, 1970) | chapter-url=http://mathunion.org/ICM/ICM1970.1/ | publisher=Gauthier-Villars | year=1971 | volume=1 | chapter=Free ideal rings and free products of rings | pages=273–278 | mr=0506389 | access-date=2010-11-26 | archive-url=https://web.archive.org/web/20171125033900/http://mathunion.org/ICM/ICM1970.1/ | archive-date=2017-11-25 | url-status=dead }}
- {{Citation | last1=Cohn | first1=P. M. | title=Free ideal rings and localization in general rings | publisher=Cambridge University Press | series=New Mathematical Monographs | isbn=978-0-521-85337-8 | mr=2246388 | year=2006 | volume=3}}
- {{Citation | last1=Cohn | first1=P. M. | title=Free rings and their relations | publisher=Academic Press | location=Boston, MA | edition=2nd | series=London Mathematical Society Monographs | isbn=978-0-12-179152-0 | mr=800091 | year=1985 | volume=19}}
- {{Citation | last1=Cohn | first1=P. M. | title=Introduction to ring theory | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Undergraduate Mathematics Series | isbn=978-1-85233-206-8 | mr=1732101 | year=2000}}
- {{eom|title=Free ideal ring}}
Further reading
- {{citation | last=Cohn | first=P.M. | authorlink=Paul Cohn | title=Skew fields. Theory of general division rings | zbl=0840.16001 | series=Encyclopedia of Mathematics and Its Applications | volume=57 | location=Cambridge | publisher=Cambridge University Press | year=1995 | isbn=0-521-43217-0 | url-access=registration | url=https://archive.org/details/skewfieldstheory0000cohn }}
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