Exercise 12, § 1

Let A be the ring of integral functions (with complex values) of a complex variable. It is an integral domain whose field of fractions is the field of meromorphic functions of a complex variable.

(a) Show that A is completely integrally closed. (b) For all $x \in A$, let $Z(x)$ be the set of zeros of $x$ (which is a closed subset of $\mathbf{C}$, which is discrete (and therefore countable) if $x \neq 0$). If $a$ is an ideal of A distinct from A and (0), show that the $Z(x)$ for $x \in a$ form a filter base $\mathscr{F}_a$. (If $x$ and $y$ are two integral functions with no zero in common, show, using the Mittag-Leffler Theorem, that it is possible to write $1/xy = u/x + v/y$, where $u$ and $v$ are in A, and deduce that $Ax + Ay = A$). Conversely, for every filter $\mathscr{F}$ on $\mathbf{C}$ to which a closed discrete subset of $\mathbf{C}$ belongs, the set $a(\mathscr{F})$ of $x \in A$ for which $Z(x) \in \mathscr{F}$ is an ideal of A. Show that $\mathscr{F}_{a(s)} = \mathscr{F}$ for every filter $\mathscr{F}$ to which a closed discrete subset of $\mathbf{C}$ belongs and $a(\mathscr{F}) \supseteq a$ for every ideal $a$ of A distinct from A and (0). (c) If $p \neq (0)$ is a prime ideal of A, show that $\mathscr{F}_p$ is an ultrafilter. (Restricting it to the case where all the sets of $\mathscr{F}_p$ are infinite, show that, if $M \in \mathscr{F}_p$ is the union of two infinite sets $M', M''$ with no point in common, one of these sets belongs to $\mathscr{F}_p$.) Conversely, for every ultrafilter $\mathscr{U}$ on $\mathbf{C}$ to which a closed discrete subset of $\mathbf{C}$ belongs, $a(\mathscr{U})$ is a maximal ideal of A; obtain the converse. (d) Let $\mathscr{U}$ be an ultrafilter on $\mathbf{C}$, the image under the canonical injection $\mathbf{Z} \to \mathbf{C}$ of a non-trivial ultrafilter on $\mathbf{Z}$. Show that there exist non-maximal prime ideals $p$ of A such that $\mathscr{F}_p = \mathscr{U}$. (For all $x \in a(\mathscr{U})$ and all $n \in \mathbf{Z}$, let $\omega_x(n)$ be the order of $x$ at the point $n$ ($\omega_x(n) = 0$ if $x(n) \neq 0$); consider the set $p$ of $x \in a(\mathscr{U})$ such that $\lim_n \omega_x = +\infty$.) (e) With the notation of (d), let m be the maximal ideal $a(\mathscr{U})$. Show that the domain $A_{m'}$ is not completely integrally closed (consider the meromorphic function $1/(\sin \pi z)$). \*