A ring R is Dedekind-finitePlanetmathPlanetmath if for a,bR, 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 productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 numberMathworldPlanetmath n such that xn=0 for every nilpotent elementMathworldPlanetmath xR

The finite dimensionality in the first example can not be extended to the infiniteMathworldPlanetmath 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).


  • 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
Canonical name Dedekindfinite
Date of creation 2013-03-22 14:18:23
Last modified on 2013-03-22 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 Neumann-finite