Exercise 8, § 2

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