RANKS OF MODULES RELATIVE TO A TORSION THEORY

Relative to a hereditary torsion theory we introduce a dimension for a module M, called τ-rank of M, which coincides with the reduced rank of M whenever is the Goldie torsion theory. It is shown that the τ-rank of M is measured by the length of certain decompositions of the τ-injective hull of M. Moreover, some relations between the τ-rank of M and complements to τ-torsionfree submodules of M are obtained.