  In this article several properties of the inverse along an element will be
studied in the context of unitary rings. New characterizations of the existence
of this inverse will be proved. Moreover, the set of all invertible elements
along a fixed element will be fully described. Futhermore, commuting inverse
along an element will be characterized. The special cases of the group inverse,
the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with
involutions) will be also considered.
