Free ideal ring
From Wikipedia, the free encyclopedia
In mathematics, especially in the field of ring theory, a (left) free ideal ring, or fir, is a ring in which all left ideals are free of unique rank. A ring such that all left ideals with at most n generators is free of unique rank is called an n-fir. A semi-fir is a ring in which all finitely generated left ideals are free of unique rank.
A commutative ring is a fir if and only if it is a principal ideal domain (a PID). Thus, the concept fir is a kind of generalisation of the concept PID to not necessarily commutative rings, and the firs partially have similar properties as PIDs; e.g., they have global homological dimension (at most) one. However, a fir is not 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). Every free ideal ring has the invariant basis number property.
[edit] References
- Cohn, P. M. (2006), Free ideal rings and localization in general rings, vol. 3, New Mathematical Monographs, Cambridge University Press, MR2246388, ISBN 978-0-521-85337-8
- Cohn, P. M. (1985), Free rings and their relations, vol. 19 (2nd ed.), London Mathematical Society Monographs, Boston, MA: Academic Press, MR800091, ISBN 978-0-12-179152-0
- Cohn, P. M. (2000), Introduction to ring theory, Springer Undergraduate Mathematics Series, Berlin, New York: Springer-Verlag, MR1732101, ISBN 978-1-85233-206-8
- Hazewinkel, Michiel, ed. (2001), “Free ideal ring”, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
