The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau({\frak a}^t)$ associated to a given ideal $\frak a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper (Lemma \ref{key lemma}), which gives a characterization of the ideal $\tau({\frak a}^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $\tau(R)$. Moreover, we prove an analogue of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Briançon-Skoda theorem."