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.{{Cite web|url=https://books.google.com/books?id=bOElBQAAQBAJ&dq=%22weakly+finite%22+%22basic+algebra%22+cohn&pg=PA108|title=Basic Algebra: Groups, Rings and Fields|first=P. M.|last=Cohn|date=December 6, 2012|publisher=Springer Science & Business Media|via=Google Books}} This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivialA trivial ring is stably finite but doesn't have IBN. 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.{{Cite web|url=https://books.google.com/books?id=u-4ADgUgpSMC&dq=%22klein%27s+nilpotence+condition%22&pg=PA23|title=Skew Fields: Theory of General Division Rings|first=Paul Moritz|last=Cohn|date=July 28, 1995|publisher=Cambridge University Press|via=Google Books}}