Generalized inverses: Uniqueness proofs and three new classes

Given any ring R with 1 and any a, b, c is an element of R, then, generalizing ideas of J. J. Koliha and P. Patricio in 2002 and of Z. Wang and J. Chen in 2012, a is called "(b, c)-pseudo-polar" if there exists an idempotent p is an element of R such that 1 - p is an element of (bR + J) boolean AND (Rc + J), pb and cp is an element of J (where J denotes the Jacobson radical of R) and p lies in the second commutant of a. This p is shown to be unique whenever it exists. A new outer generalized inverse y of a, called the (b, c)-pseudo-inverse of a, is also defined, and the existence of y is shown to imply that a is (b, c)-pseudo-polar, and hence that y is itself unique. Generalizing results of Koliha, Patricio, Wang and Chen, further connections between the (b, c)-pseudo-polar and (b, c)-pseudo-invertible properties are found, and the (b, c)-pseudo-invertibility of a(1)a(2) is shown to imply a corresponding property for a(2)a(1). Two further types of uniquely-defined outer generalized inverses are also introduced. (C) 2014 Elsevier Inc. All rights reserved.