= Solution
Take the <dual numbers> $A=k[\varepsilon]/(\varepsilon^2)$ over any <field> $k$, let $B=k$, and let $\alpha$ send $\varepsilon$ to zero. Its <kernel> is the nonzero <ideal> $(\varepsilon)$, but both <spectra of rings> have exactly one point: every <prime ideal> contains the <nilpotent element> $\varepsilon$, and the only prime of $A$ is $(\varepsilon)$. Consequently the induced map is a <homeomorphism> for the <Zariski topology>, although it is not an <isomorphism> of <schemes>:
$$
\boxed{\operatorname{Spec}k\longrightarrow\operatorname{Spec}(k[\varepsilon]/(\varepsilon^2)).}
$$
More generally, quotienting by a <nilpotent ideal> gives a <nilpotent thickening> with unchanged underlying topology. The <structure sheaf> retains information that the topology alone cannot recover.
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