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# Dedekind-finite

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).

# 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).

## Mathematics Subject Classification

16U99*no label found*

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