  This paper is concerned with ideals in a commutative Noetherian ring $R$ of
prime characteristic. The main purpose is to show that the Frobenius closures
of certain ideals of $R$ generated by regular sequences exhibit a desirable
type of `uniform' behaviour. The principal technical tool used is a result,
proved by R. Hartshorne and R. Speiser in the case where $R$ is local and
contains its residue field which is perfect, and subsequently extended to all
local rings of prime characteristic by G. Lyubeznik, about a left module over
the skew polynomial ring $R[x,f]$ (associated to $R$ and the Frobenius
homomorphism $f$, in the indeterminate $x$) that is both $x$-torsion and
Artinian over $R$.
