Complete parametrization of outer inverses in arbitrary rings

Derive a complete parametrization of all outer inverses of a fixed element in an arbitrary ring, extending the explicit descriptions available under additional structural assumptions.

Background

The paper reviews descriptions of inner and outer inverses. Whereas the set of inner inverses has an affine parametrization, the authors explain that the set of outer inverses is substantially more complicated. Explicit parametrizations are available for anti-regular rings, regular rings, idempotents, group-invertible elements, and other settings with additional structure, but the general arbitrary-ring case remains unresolved.

A complete parametrization would describe every solution of the equation xax = x for a fixed element a using a uniform algebraic expression, thereby extending the specialized results surveyed in the paper.