B-Fredholm elements in rings and algebras

In this paper, we study B-Fredholm elements in rings and algebras. After characterizing these elements in terms of generalized Fredholm elements, we will give sufficient conditions on 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 $A$.