That the monoid of all transformations of any set and the monoid of all endomorphisms of any vector space over a division ring are regular (in the sense of von Neumann) has been known for many years (see  and , respectively). A common generalization of these results to the endomorphism monoid of an independence algebra can be found in . It also follows from  that the endomorphism monoid of a free G-act is regular, where G is any group. In the present paper we use a version of the wreath product construction of ,  to determine the projective right S-acts (S any monoid) whose endomorphism monoid is regular.
Bulman-Fleming, Sydney, "Regularity and Products of Idemopotents in Endmorphism Monoids of Projective Acts" (1995). Mathematics Faculty Publications. 6.