i thought it was true that maximals ideals are always prime. and i thought prime ideals are only maximal in a PID. In this case, with have a commutative ring with unity, so I don't think this is true. But, here is how to prove that maximal ideals are prime:

let be a ring and m a maximal ideal. suppose and . then and since is maximal, This Thus . so . so m is prime. QED.