  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 by
a general element of that ideal for ideals of height at least two in a large
class of rings. Moreover, we show integral closure of sufficiently large powers
of the ideal is compatible with specialization by a general element of the
original ideal. In a polynomial ring over an infinite field, we give a class of
squarefree monomial ideals for which the integral closure is compatible with
specialization by a general linear form.
