The $cl-core$ of an ideal

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, $\mathrm{\ell }$, is equal to the *-spread and $I$ is $G_{\mathrm{\ell }}$ and weakly-$(\mathrm{\ell }-1)$-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.