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In two recent papers, the author has developed a theory of gradedannihilators of left modules over the Frobenius skew polynomial ring over acommutative Noetherian ring $R$ of prime characteristic $p$, and has shown thatthis 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 structureon the injective envelope $E$ of the simple $R$-module can be extended to astructure as a torsion-free left module over the Frobenius skew polynomialring, then $R$ is $F$-pure and has a tight closure test element. One of thecentral 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 skewpolynomial ring; a corollary is that every $F$-pure local ring of primecharacteristic, even if it is not excellent, has a tight closure test element.These results are then used, along with embedding theorems for modules over theFrobenius skew polynomial ring, to show that every excellent (not necessarilylocal) $F$-pure ring of characteristic $p$ must have a so-called `big' testelement.

An excellent $F$-pure ring of prime characteristic has a big tight closure test element