TITLE: Description of the set of prime ideals of the $R/m^2$
QUESTION [6 upvotes]: 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 ?

REPLY [3 votes]: Prime $\rm\ P = I + m^2\supset m^2\ \Rightarrow\ P\supset m\ \Rightarrow\ P = m\ $ since $\rm\ m\ $ is maximal.