The author provides examples illustrating various difficulties in extending the algebraic Fredholm theory developed in the monograph by \textit{B. A. Barnes} et al. [Riesz and Fredholm theory in Banach algebras. Boston-London-Melbourne: Pitman Advanced Publishing Program (1982; Zbl 0534.46034)] from the setting of primitive Banach algebras to that of semiprime algebras (either Banach algebras or general ones). For instance, the stability of the index under Fredholm perturbations fails in the general case. Moreover, the author describes the relation between the algebraic kernel and the pre-socle of a Banach algebra, and shows how to define a notion of rank for elements of semiprime algebras. The algebraic kernel of \(A\) is the largest ideal of algebraic elements, and the pre-socle of \(A\) is the lifting under the quotient map of the socle of \(A/\mathrm{rad}(A)\), which has been used in defining the Fredholm elements of \(A\) as a replacement for the ideal of finite rank operators.