Exercise 9, § 1

(a) Let $R$ be a ring, $A$ a subring of $R$, $x$ an element of $R$ and $S$ the set of $x^k$ ($k \geq 0$) in $R$. Show that, for $x$ to be integral over $A$, it is necessary and sufficient that $x/1 \in A[1/x]$ in the ring $S^{-1}R$. If $x$ is not integral over $A$, show that there exists a maximal ideal $m$ of $A[1/x]$ containing $1/x$ and that the inverse image of $m$ under the canonical homomorphism $A \to A[1/x]$ is a maximal ideal of $A$.

(b) Show that the integral closure $A'$ of $A$ in $R$ is the intersection of the paravaluative subrings of $R$ (Exercise 8) which contain $A$. (To see that $A'$ is contained in each of these rings, use Exercise 8 (a) and Exercise 6 (d); to see that, if $x \in R$ is not integral over $A$, there exists a subring $B$ of $R$ which is paravaluative in $R$ and such that $x \notin B$, use (a), Exercise 7 (a) and Exercise 8 (c), arguing as in Theorem 3 of no. 3.)