Let R be a commutative ring with unity and S a subset of R which is closed under multiplication. If P is a maximal element of the set of ideals which do not intersect S nontrivially, then I want to show that P is a prime ideal of R.
The Attempt at a Solution
I was trying to use a similar proof to the one that shows that every maximal ideal in R is prime. That is, given a maximal ideal M and an element xy in M consider the ideal M+(x) and show that since this ideal is either M or the whole ring, y must be in M. I'm having trouble adjusting it to work here though, if indeed this is even the correct strategy. Any sugestions?