A NOTE ON ADDITIVE RANK FUNCTIONS John A. Beachy Department of Mathematical Sciences Northern Illinois University DeKalb, IL 60115 Additive rank functions have been studied for Noetherian rings by Krause. It is shown that the notion of an additive rank function can be extended to more general classes of rings, and can be used in the characterization of semiprime Goldie rings and of orders in Artinian rings. In this paper all rings are assumed to be associative rings with an identity element, and all modules are assumed to be unital modules. The notation R–Mod is used for the category of left R-modules. The reader is referred to [1] for general terminology and notation, and to [2] for terminology and notation relating to torsion radicals and quotient categories. In [3], A. W. Goldie defined the reduced rank ρ(M ) of a finitely generated module R M over a semiprime Goldie ring R to be the uniform dimension of M/γ (M ), where γ is the Goldie torsion radical defined by the set of regular elements of R. Equivalently, ρ(M ) is given by the length of the left Q-module Q ⊗ R M , where Q is the classical ring of left quotients of R. This length is defined for any finitely generated module over Q, since Q is a semisimple Artinian ring. For a Noetherian ring R, this definition was generalized by Goldie, as follows. If N is the prime radical of R, then N is nilpotent, say N k = (0), and R/N is a semiprime Goldie ring. Thus 1