Building on previous work of Schwede, Böckle, and the author, we study test ideals by viewing them as minimal objects in a certain class of modules, called
F
F
-pure modules, over algebras of
p
−
e
p^{-e}
-linear operators. We develop the basics of a theory of
F
F
-pure modules and show an important structural result, namely that
F
F
-pure modules have finite length. This result is then linked to the existence of test ideals and leads to a simplified and generalized treatment, also allowing us to define test ideals in non-reduced settings.
Combining our approach with an observation of Anderson on the contracting property of
p
−
e
p^{-e}
-linear operators yields an elementary approach to test ideals in the case of affine
k
k
-algebras, where
k
k
is an
F
F
-finite field. As a byproduct, one obtains a short and completely elementary proof of the discreteness of the jumping numbers of test ideals in a generality that extends most cases known so far; in particular, one obtains results beyond the
Q
\mathbb {Q}
-Gorenstein case.