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
<templatestyles src="Reflist/styles.css" />
- ↑ A trivial ring is stably finite but doesn't have IBN.
Script error: No such module "Check for unknown parameters".
References
<templatestyles src="Reflist/styles.css" />
Script error: No such module "Check for unknown parameters".