# 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. 1.

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

2. 2.
3. 3.

any ring of matrices over a division ring

4. 4.

finite direct product of Dedekind-finite rings

5. 5.

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

6. 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).

## References

• 1 T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York (1991).
• 2 T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York (1999).
Title Dedekind-finite Dedekindfinite 2013-03-22 14:18:23 2013-03-22 14:18:23 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 16U99 von Neumann-finite