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
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
F
-stability number and the set of
F
F
-stable primes. We associate to every ideal
I
I
generated by a system of parameters and
x
∈
I
∗
−
I
x \in I^\ast - I
an ideal of multipliers denoted
I
(
x
)
I(x)
and obtain a family of ideals
Z
I
,
R
Z_{I,R}
. The set
Max
(
Z
I
,
R
)
\operatorname {Max}(Z_{I,R})
is independent of
I
I
and consists of finitely many prime ideals. It also equals
Max
{
P
|
P
\operatorname {Max} \{P| P
prime ideal such that
R
P
R_{P}
is
F
F
-stable
}
\}
. The maximal height of such primes defines the
F
F
-stability number.