  Let $R$ be a commutative (Noetherian) local ring of prime characteristic $p$
that is $F$-pure. This paper is concerned with comparison of three finite sets
of radical ideals of $R$, one of which is only defined in the case when $R$ is
$F$-finite (that is, is finitely generated when viewed as a module over itself
via the Frobenius homomorphism). Two of the afore-mentioned three sets have
links to tight closure, via test ideals. Among the aims of the paper are a
proof that two of the sets are equal, and a proposal for a generalization of I.
M. Aberbach's and F. Enescu's splitting prime.
