Exercise 8, § 7

Let $A$ be a ring and $a$ an ideal of $A$. Put $a^i = A$ for $i \leq 0$ and consider the $A$-algebra $\mathscr{A}(a) = \sum_{i \in \mathbf{Z}} a^{-i} v^i \subset A[v, v^{-1}]$ and the $A$-algebra $P(a) = \sum_{i \geq 0} a^i v^i \subset A[v]$. These are two graded $A$-algebras having the same total ring of fractions. Now prove that for an element $a \in A$ the following conditions are equivalent:

(i) The element $av$ of the $\mathscr{A}(a)$-algebra $A[v, v^{-1}]$ is integer over $\mathscr{A}(a)$.

(ii) The element $av$ of the $P(a)$-algebra $A[v]$ is integer over $P(a)$.

(iii) There exists an integral dependence relation:

One will say that $a$ is integer over the ideal $a$.