We introduce a new variant of tight closure associated to any fixed ideal
a
\mathfrak {a}
, which we call
a
\mathfrak {a}
-tight closure, and study various properties thereof. In our theory, the annihilator ideal
τ
(
a
)
\tau (\mathfrak {a})
of all
a
\mathfrak {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
τ
(
a
)
\tau (\mathfrak {a})
and the multiplier ideal associated to
a
\mathfrak {a}
(or, the adjoint of
a
\mathfrak {a}
in Lipman’s sense) in normal
Q
\mathbb {Q}
-Gorenstein rings reduced from characteristic zero to characteristic
p
≫
0
p \gg 0
. Also, in fixed prime characteristic, we establish some properties of
τ
(
a
)
\tau (\mathfrak {a})
similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal
τ
(
a
)
\tau (\mathfrak {a})
and the F-rationality of Rees algebras.