Exercise 11, § 1

(a) Let R be an integral domain, $(A_\alpha)$ a right directed family of subrings of R and A the union of the $\&$. Show that, if each of the $A_\alpha$ is integrally closed, so is A.

(b) Let K be a commutative field and $R = K(X, Y)$ the field of rational functions in two indeterminates over K. For every integer $n > 0$ let $A_n$ denote the subring $K[X, Y, YX^{-n}]$ of R. Show that the $A_n$ are completely integrally closed but that their union A is not. (For an element of the form $P(X, Y) . X^{-ns}$, where $P \in K[X, Y]$, to belong to $A$, show that it is necessary and sufficient that $P$ belong to $\mathfrak{g}^s$, where 3 denotes the ideal of $K[X, Y]$ generated by $X^n$ and $Y$.)