Description of the set of prime ideals of the $R/m^2$

Question

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 ?

score 10 · Accepted Answer · answered Jan 14 '11 at 10:09

10

$m/m^2$ 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 $R/m^2$.

answered Jan 14 '11 at 10:09

score 3 · Answer 2 · edited Jan 14 '11 at 19:32

3

Prime $\rm\ P = I + m^2\supset m^2\ \Rightarrow\ P\supset m\ \Rightarrow\ P = m\ $ since $\rm\ m\ $ is maximal.

edited Jan 14 '11 at 19:32

answered Jan 14 '11 at 19:19

Bill Dubuque

2 Answers2