Dedekind-finite ring explained

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.
• Any subring of a Dedekind-finite ring is Dedekind-finite.
• Any domain is Dedekind-finite.
• Any left Noetherian ring is Dedekind-finite.
• A unit-regular ring is Dedekind-finite.
• A local ring is Dedekind-finite.

See also

• Dedekind-infinite set
• Von Neumann regular ring

Notes and References

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