We introduce a new variant of tight closure associated to any fixed ideal a, which we call a-tight closure, and study various properties thereof. In our theory, the annihilator ideal tau(a) of all a-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal tau(a) and the multiplier ideal associated to a (or, the adjoint of a in Lipman's sense) in normal Q-Gorenstein rings reduced from characteristic zero to characteristic p >> 0. Also, in fixed prime characteristic, we establish some properties of tau(a) similar to those of multiplier ideals (e. g., a Briancon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal tau(a) and the F-rationality of Rees algebras.