  The purpose of this paper is to describe several applications of finiteness
properties of $F$-finite $F$-modules recently discovered by M. Hochster to the
study of Frobenius maps on injective hulls, Frobenius near-splittings and to
the nature of morphisms of $F$-finite $F$-modules.
  Among the results in the paper we show that morphisms of $F$-finite
$F$-modules have a particularly simple form, and we show that certain Frobenius
near-splittings have finitely many compatible submodules, thus generalizing a
result of M. Blickle and G. B\"ockle to the non-$F$-finite case.
