Dedekind-finite ring
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Short description: Mathematical concept
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
- ↑ 1.0 1.1 Goodearl, Kenneth (1976) (in en). Ring Theory: Nonsingular Rings and Modules. CRC Press. pp. 165–166. ISBN 978-0-8247-6354-1. https://books.google.com/books?id=5FKzb7LZMvAC&dq=%22directly+finite+rings%22&pg=PA166.
- ↑ 2.0 2.1 2.2 2.3 2.4 2.5 Lam, T. Y. (2012-12-06) (in en). A First Course in Noncommutative Rings. Springer Science & Business Media. ISBN 978-1-4684-0406-7. https://books.google.com/books?id=2PwGCAAAQBAJ.
See also
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