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Let $R$ be a (commutative Noetherian) local ring of prime characteristic thatis $F$-pure. This paper studies a certain finite set ${\mathcal I}$ of radicalideals of $R$ that is naturally defined by the injective envelope of the simple$R$-module. This set ${\mathcal I}$ contains $0$ and $R$, and is closed undertaking primary components. For a multiplicatively closed subset $S$ of $R$, theconcept of tight closure with respect to $S$, or $S$-tight closure, isdiscussed, together with associated concepts of $S$-test element and $S$-testideal. It is shown that an ideal of $R$ belongs to ${\mathcal I}$ if and onlyif it is the $S'$-test ideal of $R$ for some multiplicatively closed subset$S'$ of $R$. When $R$ is complete, ${\mathcal I}$ is also `closed under takingtest ideals', in the following sense: for each proper ideal $C$ in ${\mathcalI}$, it turns out that $R/C$ is again $F$-pure, and if $J$ and $K$ are theunique ideals of $R$ that contain $C$ and are such that $J/C$ is the (tightclosure) test ideal of $R/C$ and $K/C$ is the big test ideal of $R/C$, thenboth $J$ and $K$ belong to ${\mathcal I}$. The paper ends with severalexamples.

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