Rings with finite reduced rank

In [6], Goldie defined the reduced rank ρ(M) of a finitely generated module 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⊗RM , where Q is the classical ring of left quotients of R, which is semisimple Artinian. The reduced rank of a finitely generated module N over a Noetherian ring R was defined by Goldie as follows: if N is the prime radical of R, then N is nilpotent, say Nk = (0), and R/N is a semiprime Goldie ring, so it is possible to utilize the previous definition by considering ∑k i=1 ρ(N i−1M/N iM). The purpose of this paper is to give a general definition of reduced rank and to investigate the properties of rings which have finite reduced rank under this definition. Let R be any ring, with prime radical N , let γ denote the torsion radical cogenerated by the injective envelope E(R/N) of the left R-module