The one-sided inverse along two elements in rings.

Let R be a ring and a , b , c ∈ R . In this paper, it is shown that if a is left (b , c) -invertible and cab is regular, then a has at least one regular left (b , c) -inverse. And the expression of the (b , c) -inverse of a is given in terms of the inner inverse of cab. Moreover, the strongly left (or right) (b , c) -inverse is introduced in terms of the regularity of cab. It is shown when (a b) ∗ = a b , a is left (b , b) -invertible implies that a is strongly left (b , b) -invertible, and left (b , b) -invertibility coincides with right (b , b) -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. [ABSTRACT FROM AUTHOR]