Stably finite ring
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.[1] This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial[notes 1] 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.[2]
Notes
[edit]- ^ A trivial ring is stably finite but doesn't have IBN.
References
[edit]- ^ 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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