Dedekind-finite

A ring $R$ is Dedekind-finite if for $a,b\in R$, whenever $ab=1$ implies $ba=1$.

Of course, every commutative ring is Dedekind-finite. Therefore, the theory of Dedekind finiteness is trivial in this case. Some other examples are

1. any ring of endomorphisms over a finite dimensional vector space (over a field)

2. any division ring

3. any ring of matrices over a division ring

4. finite direct product of Dedekind-finite rings

5. by the last three examples, any semi-simple ring is Dedekind-finite.

6. any ring $R$ with the property that there is a natural number $n$ such that $x^n=0$ for every nilpotent element $x\in R$

The finite dimensionality in the first example can not be extended to the infinite case. Lam in [1] gave an example of a ring that is not Dedekind-finite arising out of the ring of endomorphisms over an infinite dimensional vector space (over a field).