Strong and weak F-regularity are equivalent for graded rings

It is shown that the tight closure of a submodule in a Artinian module is the same as its finitistic tight closure, when the modules are graded over a finitely generated N-graded ring over a perfect field. As a corollary, it is deduced that for such a graded ring, strong and weak F-regularity are equivalent. As another application, the following conjecture of Hochster and Huneke is proved: Let (R,m) be a finitely generated N-graded ring over a field with unique homogeneous maximal ideal m, then R is (weakly) F-regular if and only if R_m is (weakly) F-regular.