Left and right generalized inverses

This article examines a way to define left and right versions of the large class of "(b, c)-inverses" introduced by the writer in (2012) [6]: Given any semigroup S and any a, b, c is an element of S, then a is called left (b, c)-invertible if b is an element of Scab, and x is an element of S is called a left (b, c)-inverse of a if x is an element of Sc and xab = b, and dually c is an element of cabS, z is an element of Sb and caz = z for right (b, c)-inverses z of a. It is shown that left and right (b, c)-invertibility of a together imply (b, c)-invertibility, in which case every left (b, c)-inverse of a is also a right (b, c)-inverse, and conversely, and then all left or right (b, c)-inverses of a coincide. When b = c (e.g. for the Moore-Penrose inverse or for the pseudo-inverse of the author) left (b, b)-invertibility coincides with right (b, b)-invertibility in every strongly pi-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)-inverses. (C) 2016 Elsevier Inc. All rights reserved.