Random galois extensions of hilbertian rings

Let R be a countable Hilbertian ring with quotient field K and let L be a Galois extension of K. We generalize a result of Lior Bary-Soroker and Arno Fehm from fields to rings and prove that, for an abundance of large Galois extensions N of K within L, the integral closure of R in N is Hilbertian.