Let R be a commutative ring and let SCR is closed under multiplication. Let a to be an ideal in R such that anS=0. We further assume that if b is an ideal in R such that a Cb and a b, then bns0. (a) Prove that a is a prime ideal in R. (b) Prove that if a is an ideal satisfying the above condition for S = {1}, then a is a maximal ideal. (c) Prove that if m is a maximal ideal in R, then m satisfyies the above condition for S = {1}.