Past exam of the rings and modules course of the University of Oxford / 2023 / 2 / b / iii / Solution

ConsiderThe map sending a sequence of signs $({\epsilon }_j)$ to $({\epsilon }_j{2}^j)$ is a bijection from $\{-1,1{\}}^{{\mathbb{N}}_0}$ onto $W$. This is an uncountable set by the Cantor diagonal argument: given a proposed list $w_0,w_1,\dots$ of its elements, define $w(j)=-w_j(j)$. Then $w\in W$ differs from $w_j$ at coordinate $j$ for every $j$.

Since $\ker \pi$ is countable, $W$ cannot be contained in it. Consequently there exists