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, B of the same size over R, AB = 1 implies BA = 1. This is a stronger property for a ring than its having the invariant basis number (IBN) property. Namely, any nontrivial^[1] stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. A subring of a stably finite ring and a matrix ring over a stably finite ring is stably finite. A ring satisfying Klein's nilpotence condition is stably finite.

References

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

• P.M. Cohn (2003). Basic Algebra, Springer.