---
tag: 7VRY94A22R
kind: concept
name: Stably finite ring
msc: [16U99, 16D40]
specializes: [GV1HKEAWVN]
sources:
  - url: https://en.wikipedia.org/wiki/Dedekind-finite_ring
---

## Summary

A ring is stably finite when all its matrix rings are [[Dedekind-finite|GV1HKEAWVN]].
Being Dedekind-finite does not pass to matrix rings in general, so this is a stronger condition.
Stably finite nonzero rings have invariant basis number, so the rank of a finitely generated free module over them is well defined.

## Explanations

### L1 human

A ring is Dedekind-finite when a one-sided inverse is always two-sided.
Stable finiteness asks the same thing for square matrices of every size over the ring.
Commutative rings are stably finite: for square matrices over them, $AB = I$ gives $\det(A)\det(B) = 1$, and then $A$ is invertible.
So the rule "a left inverse of a square matrix is a right inverse" holds there in every size.