Dedekind-finite ring

In mathematics, a ring is said to be a Dedekind-finite ring if ab = 1 implies ba = 1 for any two ring elements a and b. In other words, all one-sided inverses in the ring are two-sided.

These rings have also been called directly finite rings^[1] and von Neumann finite rings.^[2]

Properties

Any finite ring is Dedekind-finite.^[2]
Any subring of a Dedekind-finite ring is Dedekind-finite.^[1]
Any domain is Dedekind-finite.^[2]
Any left Noetherian ring is Dedekind-finite.^[2]
A unit-regular ring is Dedekind-finite.^[2]
A local ring is Dedekind-finite.^[2]

References

1 2 Goodearl, Kenneth (1976). Ring Theory: Nonsingular Rings and Modules. CRC Press. pp. 165–166. ISBN 978-0-8247-6354-1.
1 2 3 4 5 6 Lam, T. Y. (2012-12-06). A First Course in Noncommutative Rings. Springer Science & Business Media. ISBN 978-1-4684-0406-7.

See also

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.