C-PRIMARY DECOMPOSITION FOR MODULES

This paper is concerned with the study of the C -primary decomposition for a module on a ring R. We use the concept of A -module, studied in [1] and [2], to define C -coprimary and C -primary modules, where C is a subset of (A) under bar = {chi({M})|M is an A-module}, see [1, Definition 1.5]. Given a proper submodule N of an R-module M, we define the C - primary decomposition of N in M. The C - primary decomposition is a generalization of the primary decomposition given by Goldman in [6]. We prove that an R- module M is a D-C - module if and only if for each proper submodule N of M, N has C - primary decomposition in M. Examples are given to illustrate the theory.