Exercise 9, § 1

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

b) Now prove that the integral closure $A'$ of A in R is the intersection of the paravaluative subrings in 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 integer over A, there exists a subring B of R, paravaluative in R and such that $x \notin B$, use a), Exercise 7 a) and Exercise 8 c), reasoning as in Theorem 3 of No. 3).