Dedekindfinite
A ring $R$ is Dedekindfinite^{} if for $a,b\in R$, whenever $ab=1$ implies $ba=1$.
Of course, every commutative ring is Dedekindfinite. 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 Dedekindfinite rings

5.
by the last three examples, any semisimple ring is Dedekindfinite.

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 Dedekindfinite arising out of the ring of endomorphisms over an infinite dimensional vector space (over a field).
References
 1 T. Y. Lam, A First Course in Noncommutative Rings, SpringerVerlag, New York (1991).
 2 T. Y. Lam, Lectures on Modules and Rings, SpringerVerlag, New York (1999).
Title  Dedekindfinite 

Canonical name  Dedekindfinite 
Date of creation  20130322 14:18:23 
Last modified on  20130322 14:18:23 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16U99 
Synonym  von Neumannfinite 