# Maximal (among non-principal ideals) Ideal Must be Prime

If $I$ is an ideal which is maximal among the ones that are not principal, then $I$ is prime.

This would mean that for all $f \in R$, $(f) \subset I$. Could I then use colon ideals? I was thinking maybe that for $P$ prime, $(P: (f) )= R$ if $f \in P$ or $(P: (f) )= P$ if $f \not\in P$.

• No, it wouldn't mean what you say it would...maximal doesn't mean that. – DonAntonio Mar 23 '14 at 12:36
• All the possible principal ideals be (f) where f is any element of R, correct? Then would it be that if I is maximal, then $I \subset (f)$ implies $(f)=R$? – math1234567 Mar 23 '14 at 18:05
• Yes @violin.lover, that last you wrote is true...if you actually meant $\;I\subset (f)\;,\;\;I\neq (f)\;$ . Still, my first comment remains... – DonAntonio Mar 23 '14 at 19:21

This time, you'll need to prove that $J = I + (f)$ is not principal if $J$ is not principal. The method of proof is pretty much the same.