Graded annihilators and tight closure test ideals

Let R be a commutative Noetherian local ring of prime characteristic p, with maximal ideal m. The main purposes of this paper are to show that if the injective envelope E of R/m has a structure as an x-torsion-free left module over the Frobenius skew polynomial ring over R (in the indeterminate x), then R has a tight closure test element (for modules) and is F-pure, and to relate the test ideal of R to the smallest ‘E-special’ ideal of R of positive height.A byproduct is an analogue of a result of Janet Cowden Vassilev: she showed, in the case where R is an F-pure homomorphic image of an F-finite regular local ring, that there exists a strictly ascending chain 0=τ0⊂τ1⊂⋯⊂τt=R of radical ideals of R such that, for each i=0,…,t−1, the reduced local ring R/τi is F-pure and its test ideal (has positive height and) is exactly τi+1/τi. This paper presents an analogous result in the case where R is complete (but not necessarily F-finite) and E has a structure as an x-torsion-free left module over the Frobenius skew polynomial ring. Whereas Cowden Vassilev's results were based on R. Fedder's criterion for F-purity, the arguments in this paper are based on the author's work on graded annihilators of left modules over the Frobenius skew polynomial ring.