Submodules of Free Modules

We have made a preliminary investigation of summands of free modules. What can we say about submodules of free modules? We have one elementary tool which can be used to analyze such submodules.

OBSERVATION 1. Let D be a submodule of the left R-module A ⊕ B, and suppose that Dπ is projective, where π is the natural projection of A ⊕ B onto B. Then D = [D ∩ A] ⊕ B′ for some submodule B’ of B isomorphic to Dπ.