TITLE: The intersection of distinct maximal ideals is not prime.
QUESTION [6 upvotes]: Let $P,Q$ be distinct maximal ideals of a ring $R$. Prove that $P\cap Q$ is not prime.

I am not sure how to prove this.  The only facts that I can think of applying are the definitions, $R/M$ is a field iff $M$ is max, and $R/P$ is an ID iff $P$ is prime.
Any suggestions would be appreciated.

REPLY [4 votes]: Hint $\rm\: R/(I\cap J)\cong R/I \times R/J\:$ has obvious zero divisors $\rm (\bar i,0)(0,\bar j)=(0,0)\:$ if $\rm\:I\not\subset J, J\not\subset I$
Note that the other proofs reduce to this by factoring out $\rm\:P\cap Q.$ This ubiquitous modular reduction technique is better learned sooner than later (hence its emphasis in said link by the master algebraist Irving Kaplansky very early in his textbook).