  In this note we introduce and study basic properties of two types of modules
over a commutative noetherian ring $R$ of positive prime characteristic. The
first is the category of modules of finite $F$-type. These objects include
reflexive ideals representing torsion elements in the divisor class group of
$R$. The second class is what we call $F$-abundant modules. These include, for
example, the ring $R$ itself and the canonical module when $R$ has positive
splitting dimension. We prove various facts about these two categories and how
they are related, for example that $\mathrm{Hom}_R(M,N)$ is maximal
Cohen-Macaulay when $M$ is of finite $F$-type and $N$ is $F$-abundant, plus
some extra (but necessary) conditions. Our methods allow us to extend previous
results by Patakfalvi-Schwede, Yao and Watanabe. They also afford a deeper
understanding of these objects, including complete classifications in many
cases of interest, such as complete intersections and invariant subrings.
