  Over a Cohen-Macaulay ring we consider two extensions of the maximal
Cohen-Macaulay modules from the viewpoint of definable subcategories, which are
closed under direct limits, direct products and pure submodules. After
describing these categories, we compare them and consider which properties they
inherit from the maximal Cohen-Macaulay modules. We then consider some further
properties of these classes and how they interact with the entire module
category.
