Abstract
Let be a ring and . In this paper, it is shown that if a is left -invertible and cab is regular, then a has at least one regular left -inverse. And the expression of the -inverse of a is given in terms of the inner inverse of cab. Moreover, the strongly left (or right) -inverse is introduced in terms of the regularity of cab. It is shown when , a is left -invertible implies that a is strongly left -invertible, and left -invertibility coincides with right -invertibility. As applications, we consider when the one-sided core inverse is core invertible. It is shown that left core invertibility coincides with right core invertibility in every strongly π-regular ring.
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