Unit-regular ring

Summary

A ring $R$ is unit-regular if every element $a$ has a unit $u$ with $aua = a$. This strengthens von Neumann regularity, where $u$ is only required to be some element of $R$. Unit-regular rings are Dedekind-finite, and for a vector space $V$ the ring of endomorphisms is unit-regular exactly when $V$ is finite-dimensional. The notion matters in the study of cancellation of modules and direct sum decompositions.