Let $R$ be a commutative ring and $m\subseteq R$ be a maximal ideal. Can you describe the set of prime ideals of the $R/m^2$. Are they all maximal ?
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$m/m^2$ is a maximal ideal of $R/m^2$, and it is nilpotent; it is therefore the nilradical, and being prime, is the only prime ideal of $R/m^2$.
Prime $\rm\ P = I + m^2\supset m^2\ \Rightarrow\ P\supset m\ \Rightarrow\ P = m\ $ since $\rm\ m\ $ is maximal.