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.