In this paper the global
dimension of any complete, wellpowered abelian category wilh injective envelopes
in calculated relative to the torsion theory of A. W. Goldie and is found
to be always one or zero. The rings R such that the left module category
Rℳ has global dimension zero are precisely those such that every module
having zero singular submodule is injective. These rings are characterized as
being of the form T ⊕ S (ring direct sum) where T is a ring having essential
singular ideal and S is semi-simple with minimum condition. The rings with
essential singular ideal are precisely those which are torsion as left modules over
themselves.
