The $cl-core$ of an ideal

TL;DR

This paper generalizes the concept of core to $cl$-core for Nakayama closures, especially tight closure, and explores conditions under which the core and *-core coincide or differ, with applications to specific rings.

Contribution

It introduces the $cl$-core for Nakayama closures, analyzes the equality of core and *-core in various rings, and extends the notion of minimal reductions to *-reductions.

Findings

core and *-core coincide in certain one-dimensional domains

*- core of tightly closed ideals in some semigroup rings is tightly closed

generalized minimal reductions to *-reductions

Abstract

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.