Let be an integral domain and let and be relatively prime integers greater than 0. Prove that the ideal is prime in .
We will show that is an integral domain by demonstrating that it is isomorphic to a subring of the integral domain . To that end, define a ring homomorphism by extending , , for homomorphically. Certainly then , so that .
Now let , and consider the coset . We may eliminate powers of greater than in favor of powers of in the representative , so that in fact , where . Thus every element of has the form , where . Suppose now that . Then we have . Now letting , we have .
We claim that the exponents are pairwise distinct. To see this, suppose that . Mod , we have . Since is relatively prime to , it is a unit mod . Thus . However, since , this yields . Then , so that . Thus we have for all , so for all , and thus .
So , and by the first isomorphism theorem for rings, is isomorphic to a subring of the integral domain . Thus is an integral domain, and so the ideal is prime.