  In two recent papers, the author has developed a theory of graded
annihilators of left modules over the Frobenius skew polynomial ring over a
commutative Noetherian ring $R$ of prime characteristic $p$, and has shown that
this theory is relevant to the theory of test elements in tight closure theory.
One result of that work was that, if $R$ is local and the $R$-module structure
on the injective envelope $E$ of the simple $R$-module can be extended to a
structure as a torsion-free left module over the Frobenius skew polynomial
ring, then $R$ is $F$-pure and has a tight closure test element. One of the
central results of this paper is the converse, namely that, if $R$ is $F$-pure,
then $E$ has a structure as a torsion-free left module over the Frobenius skew
polynomial ring; a corollary is that every $F$-pure local ring of prime
characteristic, even if it is not excellent, has a tight closure test element.
These results are then used, along with embedding theorems for modules over the
Frobenius skew polynomial ring, to show that every excellent (not necessarily
local) $F$-pure ring of characteristic $p$ must have a so-called `big' test
element.
