Generalized inverses: Uniqueness proofs and three new classes

Given any ring R   with 1 and any a,b,c∈Ra,b,c∈R, then, generalizing ideas of J.J. Koliha and P. Patrício in 2002 and of Z. Wang and J. Chen in 2012, a   is called “(b,c)(b,c)-pseudo-polar” if there exists an idempotent p∈Rp∈R such that 1−p∈(bR+J)∩(Rc+J)1−p∈(bR+J)∩(Rc+J), pb and cp∈Jpb and cp∈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)(b,c)-pseudo-inverse of a, is also defined, and the existence of y is shown to imply that a   is (b,c)(b,c)-pseudo-polar, and hence that y   is itself unique. Generalizing results of Koliha, Patrício, Wang and Chen, further connections between the (b,c)(b,c)-pseudo-polar and (b,c)(b,c)-pseudo-invertible properties are found, and the (b,c)(b,c)-pseudo-invertibility of a1a2a1a2 is shown to imply a corresponding property for a2a1a2a1. Two further types of uniquely-defined outer generalized inverses are also introduced.