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 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.