The characterizations of one-sided generalized inverses

AbstractIn this article, some characterizations of one-sided generalized inverses are investigated. Let R be a *-ring and a∈R. It is shown that for k≥2, a* is right (ak,a) invertible if and only if a∈a2R∩R†. In particular, the expression of a† are given whenever a∈a2R∩R†. Moreover, it is also proven that a∈a2R∩R† if and only if (a∗a)n is right invertible along a if and only if a∗an is right invertible along a, where n is an arbitrary positive integer. Finally, we present some characterizations of a∈a2R∩R† by using projections and units.