Generalized invertibility in two semigroups of a ring

In [Linear Multilinear Algebra 43 (1997) 137], Puystjens and Hartwig proved that given a regular element t of a ring R with unity 1, then t has a group inverse if and only if u=t2t−+1−tt− is invertible in R if and only if v=t−t2+1−t−t is invertible in R. There, Hartwig posed the pertinent question whether the inverse of u and v could be directly related. Similar equivalences appear in the characterization of Moore–Penrose and Drazin invertibility, and therefore analogous questions arise. We present a unifying result to answer these questions not only involving classical invertibility, but also some generalized inverses as well.