A note on the Matlis category equivalence

We provide additional information concerning the important Matlis category equivalence between the categories of h-divisible torsion and R-complete torsion-free R-modules over any domain R. We identify the h-divisible modules that correspond under this category equivalence to the Enochs cotorsion torsion-free modules as the weak-injective modules. As a consequence, we can characterize weak-injective modules in terms of their flat covers. We also determine the range of the injective dimensions of pure-injective modules.