(In contrast to the usual convention in dealing with a torsion pair or a "torsion theory", we name first the torsion-free class, then the torsion class: this fits to the rule that in a rough thought, we visualise maps as going from left to right, and a torsion pair concerns regions with "no maps backwards".)
If (F,T) is a torsion pair in mod A, then any A-module M has a largest submodule tM which belongs to T. This submodule tM is called the torsion-submodule of M. Note that M/tM belongs to F and is the largest factor module of M which belongs to F.