Let be a principal ideal domain, let be nonzero, and let . Given a prime , say , where does not divide . Prove that is module-isomorphic to if and to if .
We begin with some lemmas.
Lemma 1: Let be ideals of a ring , and consider as an -module. Then . Proof: If , then . If , then . Thus .
Lemma 2: Let be a principal ideal domain and let be nonzero with . Note that . Prove that . Proof: Let be given by . (This is well-defined since is a domain, and is clearly an -module homomorphism.) Certainly is surjective. Now if , then , so that . Conversely, if , then for some , and so . By the First Isomorphism Theorem, .
Lemma 3: Let be a principal ideal domain and such that . If , then . Proof: Say . Note that for some , so that . Now , so that .
If , then , so that . By Lemma 1, we have by the Third Isomorphism Theorem. Using Lemma 2, this is isomorphic to .
If with (say) , then , using Lemma 3 and the Third Isomorphism Theorem.