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]