  In this paper, we first prove that the absorption law for one-sided inverses
along an element holds, deriving the absorption law for the inverse along an
element. We then apply this result to obtain the absorption law for the inverse
along different elements. Also, the reverse order law and the existence
criterion for the inverse along an element are given by centralizers in a ring.
Finally, we characterize the Moore-Penrose inverse by one-sided invertibilities
in a ring with involution.
