Exercise 17, § 9

Let $(\mathbf{K}_i)_{i \in I}$ be a family of fields. For every subset $J$ of $I$, let $e_J$ denote the element of the product $K = \prod_{i \in I} \mathbf{K}_i$ whose $i$-th component is equal to 0 if $i \in J$ and to 1 if $i \in I - J$.

(a) Let $x = (x_i)$ be an element of $K$ and $J(x)$ the set of elements $i \in I$ such that $x_i = 0$. Show that there exists an invertible element $u$ in $K$ such that $x = ue_{J(x)} = e_{J(x)}u$.

(b) Let $a$ be a left (resp. right) ideal of $K$ distinct from $K$. Let $\mathscr{F}_a$ be the set of subsets $J$ of $I$ such that $e_J \in a$. Show that $\mathscr{F}_a$ has the following properties: $(b_1)$ If $J \in \mathscr{F}_a$ and $J' \supset J$, then $J' \in \mathscr{F}_a$. $(b_2)$ If $J_1, J_2 \in \mathscr{F}_a$, then $J_1 \cap J_2 \in \mathscr{F}_a$. $(b_3)$ $\varnothing \notin \mathscr{F}_a$. Show that $x \in a$ is equivalent to $J(x) \in \mathscr{F}_a$ (use (a)). Deduce that $a$ is a two-sided ideal.

(c) A set of subsets of $I$ is called a filter if it satisfies properties $(b_1), (b_2), (b_3)$ above (cf. General Topology, I, § 6). Show that the mapping $a \mapsto \mathscr{F}_a$ is a strictly increasing bijection of the set of ideals of $K$ distinct from $K$ onto the set of filters of $I$. Show that $a$ is maximal if and only if $\mathscr{F}_a$ is an ultrafilter.

(d) Let $\mathscr{F}$ be a non-trivial ultrafilter on the set $P$ of prime numbers, let $a$ be the corresponding ideal of the ring $K = \prod_{p \in P} \mathbf{Z}/p\mathbf{Z}$ and let $k = K/a$. Show that $k$ is a field of characteristic 0.