  This paper studies the jumping coefficients of principal ideals of regular
local rings.
  Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of
essentially finite type over a field and $F$-finite, bounded intervals contain
finitely many jumping coefficients and that those are rational. In a later
paper they extended these results to principal ideals of $F$-finite complete
regular local rings. The aim of this paper is to extend these results on the
discreteness and rationality of jumping coefficients to principal ideals of
arbitrary (i.e. not necessarily $F$-finite) excellent regular local rings
containing fields of positive characteristic.
  Our proof uses a very different method: we do not use $D$-modules and instead
we analyze the modules of nilpotents elements in the injective hull or $R$
under some non-standard Frobenius actions. This new method undoubtedly holds a
potential for more applications.
