Dedekind-finite ring
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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
- If a ring is finite, then it is Dedekind-finite.[2]
- Any subring of a Dedekind-finite ring is Dedekind-finite.[1]
- If a ring is a domain then it 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
- ^ a b Goodearl, Kenneth (1976). Ring Theory: Nonsingular Rings and Modules. CRC Press. pp. 165–166. ISBN 978-0-8247-6354-1.
- ^ a b c d e f Lam, T. Y. (2012-12-06). A First Course in Noncommutative Rings. Springer Science & Business Media. ISBN 978-1-4684-0406-7.