Let be a commutative (unitary) ring. An ideal is called a prime ideal iff .Now, suppose that Q is a prime ideal of R such that . Pick , so then such that So I need to show that ... How should I start?
Lemma: If is a prime ideal and then or or ... or .
Proof: The proof is by induction. If that is true because that is the definition of what it means being a prime ideal. So say it is true for . Now say then it means then by property of being a prime ideal means or . By the inductive step we get or ... or or . And that completes the proof.
Therefore given and ideal containing let then it means for some . However, by above.