Let P be a finitely
generated projective right A-module with trace ideal T and A-endomorphism ring
B. Associated with P are the TTF classes, 𝒯F= {⋅AX∣P ⊗ X = 0} and
𝒯H= {XA∣Hom(P,X) = 0}. An investigation of these TTF classes yields
characterizations of various conditions on P and T; e.g., (1) ⋅BP is projective (flat)
and (2) ⋅AT is projective (flat). The concept of weak stability for a hereditary torsion
class is introduced and characterizations are given.