In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying Klein's nilpotence condition is stably finite.
Notes
- A trivial ring is stably finite but doesn't have IBN.
References
- Cohn, P. M. (December 6, 2012). "Basic Algebra: Groups, Rings and Fields". Springer Science & Business Media – via Google Books.
- Cohn, Paul Moritz (July 28, 1995). "Skew Fields: Theory of General Division Rings". Cambridge University Press – via Google Books.
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