we first need an important fact:
Fact: every ideal of is in the form where each is an ideal of
Proof: it's clear that is an ideal of conversely, suppose is any ideal of for any consider the map where and are the inclusion and the projection maps
respectively. let see that is an ideal of and Q.E.D.
now let be any ideal of we have we know that is prime iff is a domain. now suppose for some choose and
let and then and so in this case is not a domain. thus in order for
to be a domain, we must have for all but one which we'll call it then and clearly is a prime ideal of iff is a prime ideal of