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
n a 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
$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$
such that
F0,...,
Fn?1 are finitely generated free and
Kn?1 is
T-finitely generated; a left
R-module
M is called (
T,
n)-injective, if Ext
n R (
A,
M) = 0 for each (
T,
n+1)-presented left
R-module
A; a right R-module M is called (
T,
n)-flat, if Tor
R n (
M,
A) = 0 for each (
T,
n+1)-presented left
R-module A. A ring R is called (
T,
n)-coherent, if every (
T,
n+1)-presented module is (
n + 1)-presented. Some characterizations and properties of these modules and rings are given.