Exercise 9, § 7

(a) Let A and a be as above and let b be an ideal of A containing a. Now prove that the following conditions are equivalent: (i) Every element of b is integral over a. (ii) The $\mathscr{A}(a)$-graded algebra $\mathscr{A}(b)$ is integral. (iii) The P(a)-graded algebra P(b) is integral. (iv) There exists an integer $k \geqslant 0$ such that $a.b^k = b^{k+1}$ and hence $a^n.b^k = b^{k+n}$ for all positive integers n. We say that b is integral over a. (b) Now prove that there exists a greatest ideal among those which are integral over a. It is denoted $\overline{a}$ and is called the integral closure of a in A. Verify that if $a \subset b$ then $\overline{a} \subset \overline{b}$ and that $\overline{a} = \overline{a}$. Now prove that if the integral closure $\overline{P(a)}$ is a P(a)-finitely generated module, then $a.a^k = a^{k+1}$ for k sufficiently large. (c) Now prove that if $a \in A$ satisfies the integral dependence relation

then $a.(a + Aa)^{k-1} = (a + Aa)^k$. Deduce that the integral closure of a in A coincides with the set of elements of A which are integral over a.