Exercise 11, § 9

Let $P$ be the set of prime numbers and $A$ the product ring of the fields $\mathbf{Z}/p\mathbf{Z}$, $p \in P$. Let $a$ be the subset of $A$ consisting of the elements $(a_p)_{p \in P}$ such that $a_p \neq 0$ only for a finite number of indices $p$.

(a) $a$ is an ideal of $A$.

(b) Let $B = A/a$. For every integer $n > 0$ and every $b \neq 0$ in $B$, there exists one and only one element $b'$ of $B$ such that $nb' = b$ (note that if $p \in P$ does not divide $n$, multiplication by $n$ in $\mathbf{Z}/p\mathbf{Z}$ is bijective). Deduce that B contains a subfield isomorphic to $\mathbf{Q}$.