  Continuing ideas of a recent preprint of Schwede arXiv:0906.4313 we study
test ideals by viewing them as minimal objects in a certain class of $F$-pure
modules over algebras of p^{-e}-linear operators. This shift in the viewpoint
leads to a simplified and generalized treatment, also allowing us to define
test ideals in non-reduced settings.
  In combining this with an observation of Anderson on the contracting property
of p^{-e}-linear operators we obtain an elementary approach to test ideals in
the case of affine k-algebras, where k is an F-finite field. It also yields a
short and completely elementary proof of the discreteness of their jumping
numbers extending most cases where the discreteness of jumping numbers was
shown in arXiv:0906.4679.
