Let be a finitely generated module over a principal ideal domain . Describe the structure of .
We begin with a lemma.
Lemma: Let be a domain, and let and be (left, unital) -modules. Then . Proof: Let . Then there exists a nonzero element such that . Then , and so and . So and . Let . Then there exist nonzero elements such that and . Now is nonzero in (since it is a domain), and we have . So .
Now by the Fundamental Theorem of Finitely Generated Modules over a PID (FTFGMPID), if is such a module, we have for some primes and natural numbers . Now , since is torsion-free (since is a domain) while… the stuff inside the big sum is all torsion.
Using this previous exercise, we have that .