  We expand the notion of core to $cl$-core for Nakayama closures $cl$. In the
characteristic $p>0$ setting, when $cl$ is the tight closure, denoted by *, we
give some examples of ideals when the core and the *-core differ. We note that
*-core$(I)=$ core$(I)$, if $I$ is an ideal in a one-dimensional domain with
infinite residue field or if $I$ is an ideal generated by a system of
parameters in any Noetherian ring. More generally, we show the same result in a
Cohen--Macaulay normal local domain with infinite perfect residue field, if the
analytic spread, $\ell$, is equal to the *-spread and $I$ is $G_{\ell}$ and
weakly-$(\ell-1)$-residually $S_2$. This last is dependent on our result that
generalizes the notion of general minimal reductions to general minimal
*-reductions. We also determine that the *-core of a tightly closed ideal in
certain one-dimensional semigroup rings is tightly closed and therefore
integrally closed.
