I have to show that for a commutative ring with unity, every prime ideal is a maximal ideal.
I'm having trouble. My idea was to let M be a prime ideal. Assume M is not maximal, that is, assume there exists a proper ideal N in R so N contains M, and then derive a contradiction.
I'm getting nowhere with this idea though... Am I on the wrong track?
ANY help would be greatly apprectiated. Thanks. :-)