M is called a dimension
module if d(A + B) = d(A) + d(B) − d(A ∩ B) holds for all submodules
A and B of M, where d(M) denotes the Goldie (uniform) dimension of a
module M. We characterize these modules as the modules which have no
submodules of the form X ⊕ X∕Y with Y an essential submodule of X. As a test,
the structure of a completely decomposable injective dimension module is
determined.
