Stably finite ring
From HandWiki
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
- ↑ 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. https://books.google.com/books?id=bOElBQAAQBAJ&dq=%22weakly+finite%22+%22basic+algebra%22+cohn&pg=PA108.
- ↑ Cohn, Paul Moritz (July 28, 1995). "Skew Fields: Theory of General Division Rings". Cambridge University Press. https://books.google.com/books?id=u-4ADgUgpSMC&dq=%22klein%27s+nilpotence+condition%22&pg=PA23.
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