Specialization and integral closure

I ′/(x) = I ′/(x) , where x = ∑n i=1 ziai is a generic element for I defined over the polynomial ring R ′ = R[z1, . . . , zn] and I ′ denotes the extension of I to R. This result can be paraphrased by saying that an element is integral over I if it is integral modulo a generic element of the ideal. Other, essentially unrelated, results about lifting integral dependence have been proved by Teissier, Gaffney and Kleiman, and Gaffney ([23], [24] [6], [5, 5.4]). They go under the name ‘principle of specialization of integral dependence’ and play an important role in equisingularity theory. The above theorem opens the possibility for proofs using induction on the height. This yields, for instance, a quick proof of Huneke’s and Itoh’s celebrated result on integral closures