Tight closure with respect to a multiplicatively closed subset of an F-pure local ring

Let R be a (commutative Noetherian) local ring of prime characteristic that is F  -pure. This paper studies a certain finite set II of radical ideals of R that is naturally defined by the injective envelope E of the simple R  -module. This set II contains 0 and R, and is closed under taking primary components. For a multiplicatively closed subset S of R, the concept of tight closure with respect to S, or S-tight closure, is discussed, together with associated concepts of S-test element and S  -test ideal. It is shown that an ideal aa of R   belongs to II if and only if it is the S′S′-test ideal of R   for some multiplicatively closed subset S′S′ of R. When R   is complete, II is also ‘closed under taking test ideals’, in the following sense: for each proper ideal cc in II, it turns out that R/cR/c is again F  -pure, and if gg and hh are the unique ideals of R   that contain cc and are such that g/cg/c is the (tight closure) test ideal of R/cR/c and h/ch/c is the big test ideal of R/cR/c, then both gg and hh belong to II. The paper ends with several examples.