  We prove that the integral closedness of any ideal of height at least two is
compatible with specialization by a generic element. This opens the possibility
for proofs using induction on the height of an ideal. Also, with additional
assumptions, we show that an element is integral over a module if it is
integral modulo a generic element of the module. This turns questions about
integral closures of modules into problems about integral closures of ideals,
by means of a construction known as Bourbaki ideal.
