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Let (R,m) be an equidimensional excellent local ring of prime characteristic p. We give an alternate proof of the existence of a test exponent for any given c ∈ R◦ and all ideals generated by (full or partial) systems of parameters. This follows from a more general result about the existence of a test exponent for any given Artinian R-module. If we further assume R is Cohen-Macaulay, then we show that there exists a test exponent for any given c ∈ R◦ and all finite length modules with finite projective dimension. We then apply the results to F -rational rings and relate the results to a new kind of F -rational signature of R.

Test exponents for modules with finite phantom projective dimension