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The cl-core of an ideal

Published online by Cambridge University Press:  03 June 2010

Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003–8001, U.S.A. e-mail:
Department of Mathematics, University of New Mexico, Albuquerque, NM 87131–0001, U.S.A. e-mail:


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, ℓ, is equal to the *-spread and I is G and weakly-(ℓ − 1)-residually S2. 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.

Research Article
Copyright © Cambridge Philosophical Society 2010

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