Exercise 29, § 4

Let A be an integrally closed Nagata ring, and x an element of the field of fractions of A, not belonging to A. Let B be the ring A[x] and a, b two nonzero elements of A such that bx = a.

a) Show that the ring $B\left[\frac{1}{b}\right]$ is integrally closed.

b) Show that for every maximal ideal $m$ of $B$, the integral closure of $B_m$ is a finite $B_m$-algebra. (Use the preceding exercise.) c) Show that the integral closure of $B$ is a finite $B$-algebra. (Use Exercise 26.)