$F$-injective rings and $F$-stable primes

The notion of stability of the highest local cohomology module with respect to the Frobenius functor originates in the work of R. Hartshorne and R. Speiser. R. Fedder and K.-i. Watanabe examined this concept for isolated singularities by relating it to $F$-rationality. The purpose of this note is to study what happens in the case of non-isolated singularities and to show how this stability concept encapsulates a few of the subtleties of tight closure theory. Our study can be seen as a generalization of the work by Fedder and Watanabe. We introduce two new ring invariants, the $F$-stability number and the set of $F$-stable primes. We associate to every ideal $I$ generated by a system of parameters and $x \in I^\ast - I$ an ideal of multipliers denoted $I(x)$ and obtain a family of ideals $Z_{I,R}$. The set $\operatorname{Max}(Z_{I,R})$ is independent of $I$ and consists of finitely many prime ideals. It also equals $\operatorname{Max} \{P\vert P$ prime ideal such that $R_{P}$ is $F$-stable$\}$. The maximal height of such primes defines the $F$-stability number.