Exercise 17, § 1

Let K be a field, B the polynomial ring K[X_n]_{n ∈ $\mathbf{N}$} and A the subring of B generated by the monomials X_n^2 and X_n^3 for all n ∈ $\mathbf{N}$; show that B is the integral closure of A and that the conductor of B in A is reduced to 0; but, if S = A − {0}, the conductor of S^{-1}B in S^{-1}A is not reduced to 0 and S^{-1}A is integrally closed.