Units and one-sided units in regular rings

A ring R is unit regular if for every $a \in R$, there is a unit $x \in R$ such that $axa = a$, and one-sided unit regular if for every $a \in R$, there is a right or left invertible element $x \in R$ such that $axa = a$. In this paper, unit regularity and one-sided unit regularity are characterized within the lattice of principal right ideals of a regular ring R (Theorem 3). If M is an A-module and $R = {\text{End}_A}$ M is a regular ring, then R is unit regular if and only if complements of isomorphic summands of M are isomorphic, and R is one-sided unit regular if and only if complements of isomorphic summands of M are comparable with respect to the relation ``is isomorphic to a submodule of'' (Theorem 2). A class of modules is given for whose endomorphism rings it is the case that regularity in conjunction with von Neumann finiteness is equivalent to unit regularity. This class includes all abelian torsion groups and all nonreduced abelian groups with regular endomorphism rings.