Relative test elements for tight closure

Test ideals play a crucial role in the theory of tight closure developed by Melvin Hochster and Craig Huneke. Recently, Karen Smith showed that test ideals are closely related to certain multiplier ideals that arise in vanishing theorems in algebraic geometry. In this paper we develop a generalization of the notion of test ideals: for complete local rings R and S, where S is a module-finite extension of R, we define a module of relative test elements T(S,R) which is a submodule of HomR(S,R).