Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed
under homomorphic images and extensions. Let T be a weak torsion class of left R-modules
and na positive integer. Then a left R-module M is called T-finitely generated if there exists a
finitely generated submodule N such that M/N∈ T; a left R-module A is called (T, n)-
presented if there exists an exact sequence of left R-modules 0 → K_ n-1 → F_ n-1 → ⋯ →
F_1 → F_0 → M → 0 0→ K n− 1→ F n− 1→⋯→ F 1→ F 0→ M→ 0 such that F 0,..., F n− 1 are …