  In this paper, we study B-Fredholm elements in rings and algebras. After
characterising these elements in terms of generalized Fredholm elements, we
will give a condition on the socle of a unital primitive Banach algebra $A$,
under which we prove that an element of $A$ is a B-Fredholm element of index
$0$ if and only if it is the sum of a Drazin invertible element of $A$ and an
element of the socle of $A$.
