|
|
|
|
Dedekind-finite
|
(Definition)
|
|
|
A ring  is Dedekind-finite if for
 , whenever  implies  .
Of course, every commutative ring is Dedekind-finite. Therefore, the theory of Dedekind finiteness is trivial in this case. Some other examples are
- any ring of endomorphisms over a finite dimensional vector space (over a field)
- any division ring
- any ring of matrices over a division ring
- finite direct product of Dedekind-finite rings
- by the last three examples, any semi-simple ring is Dedekind-finite.
- any ring
 with the property that there is a natural number  such that  for every nilpotent element 
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).
- 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).
|
"Dedekind-finite" is owned by CWoo.
|
|
(view preamble)
| Other names: |
von Neumann-finite |
|
|
Cross-references: infinite dimensional, infinite, nilpotent element, natural number, property, semi-simple ring, direct product, finite, matrices, division ring, field, vector space, finite dimensional, ring of endomorphisms, theory, commutative ring, implies, ring
There are 2 references to this entry.
This is version 7 of Dedekind-finite, born on 2004-04-15, modified 2007-07-02.
Object id is 5766, canonical name is DedekindFinite.
Accessed 2223 times total.
Classification:
| AMS MSC: | 16U99 (Associative rings and algebras :: Conditions on elements :: Miscellaneous) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
