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

1. A trivial ring is stably finite but doesn't have IBN.