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In this paper, we prove a result similar to results of Itoh and Hong-Ulrich,proving that integral closure of an ideal is compatible with specialization bya general element of that ideal for ideals of height at least two in a largeclass of rings. Moreover, we show integral closure of sufficiently large powersof the ideal is compatible with specialization by a general element of theoriginal ideal. In a polynomial ring over an infinite field, we give a class ofsquarefree monomial ideals for which the integral closure is compatible withspecialization by a general linear form.

Specialization of Integral Closure of Ideals by General Elements