Here is one of those problems that invariably shows up in homework for first course on ring theory: Assume and are commutative rings with 1. Prove that if is a ring homomorphism, and is a prime ideal in , then is a prime ideal in .
Showing that is an ideal in is routine and I shall omit the details. To show that is a prime ideal in , one way to proceed is to apply the definition of a prime ideal: if , then we want to show that and . I think this is the kind of approach that seems natural for a beginner in ring theory (like me). But recently, reading “Introduction to Commutative Algebra” by Atiyah & Macdonald, I learnt the following solution, which seems more insightful. The idea is as follows:
To show that is a prime ideal in , it suffices to prove that is an integral domain. We claim that is isomorphic to a subring of , and this would finish the proof immediately for the following reason: We know is an integral domain (because is a prime ideal in ) and every subring of an integral domain is again an integral domain. So we need to exhibit an injective ring homomorphism . This map is explicitly given by
For every , we have
which shows that is a ring homomorphism. We will now show that is injective. Assume for some . Then, by definition,
which implies that so that . Thus, is injective. We have proved that is an injective ring homomorphism. As a result, is isomorphic to a subring of , and so is an integral domain, proving that is prime ideal in .