A pair (F,T) of full subcategories of mod A is said to be a torsion pair provided
  1. HomA(T,F) = 0 for all F in F, T in T,
  2. Maximality of F: If M is an A-module with HomA(M,F) = 0 for all F in F, then M belongs to T.
  3. Maximality of T: If M is an A-module with HomA(T,M) = 0 for all T in T, then M belongs to F.
If such a torsion pair is given, the modules in T are said to be the torsion modules, those in F the torsionfree modules.

(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.