This paper is concerned with the study of the

-primary decomposition for a module on a ring R. We use the concept of

-module, studied in [1] and [2], to define

-coprimary and

-primary modules, where

is a subset of

see [1, Definition 1.5]. Given a proper submodule N of an R-module M, we define the -primary decomposition of N in M. The -primary decomposition is a generalization of the primary decomposition given by Goldman in [6]. We prove that an R-module M is a -module if and only if for each proper submodule N of M, N has -primary decomposition in M. Examples are given to illustrate the theory.

torsion theory, primary decomposition, lattice.