R will be a ring with identity and module M will be unital left R-modules. In this paper, we introduce the concept of modules having the pure intersection property (the PIP). We investigate the properties of modules with the PIP. We give a characterization of modules with the PIP, among others and prove that for a flat module M, M has the PIP if and only if for any pure submodules N and L, N + L is flat.

pure submodule, flat module, projective module, prime module, torsion-free module.