Left and right generalized inverses

This article examines a way to define left and right versions of the large class of “(b,c)(b,c)-inverses” introduced by the writer in (2012) [6]: Given any semigroup S   and any a,b,c∈Sa,b,c∈S, then a is called left  (b,c)(b,c)-invertible   if b∈Scabb∈Scab, and x∈Sx∈S is called a left  (b,c)(b,c)-inverse of a   if x∈Scx∈Sc and xab=bxab=b, and dually c∈cabSc∈cabS, z∈Sbz∈Sb and caz=zcaz=z for right (b,c)(b,c)-inverses z of a  . It is shown that left and right (b,c)(b,c)-invertibility of a   together imply (b,c)(b,c)-invertibility, in which case every left (b,c)(b,c)-inverse of a   is also a right (b,c)(b,c)-inverse, and conversely, and then all left or right (b,c)(b,c)-inverses of a   coincide. When b=cb=c (e.g. for the Moore-Penrose inverse or for the pseudo-inverse of the author) left (b,b)(b,b)-invertibility coincides with right (b,b)(b,b)-invertibility in every strongly π-regular semigroup. A fundamental result of Vaserstein and Goodearl, which guarantees the left-right symmetry of Bass's property of stable range 1, is extended from two-sided inverses to left or right inverses, and, for central b  , to left or right (b,b)(b,b)-inverses.