We characterize the non-modular models of a unidimensional first-order theory of modules as the elementary submodels of its prime pure-injective model. We show that in case the maximal non-modular submodel of a given model splits off this is true for every such submodel, and we thus obtain a cancellation result for this situation. Although the theories in question always have models (in every big enough power) whose maximal non-modular submodel do split off, they may as well have others where they don't. We present a corresponding example.
