Exercise 8, § 2

Let $B$ be the additive group $\mathbf{Z} \oplus (\mathbf{Z}/2\mathbf{Z})$. We endow it with a commutative ring structure by setting $(a, b).(a', b') = (aa', ab' + ba')$ for $(a, b) \in B$ and $(a', b') \in B$. Let $\varepsilon = (0, 1) \in B$. Let $f(X) = \varepsilon X(X - 1) \in B[X]$. Then $f(\alpha) = 0$ for every $\alpha \in B$.