Let be a principal ideal domain, let be a torsion (left, unital) -module, and let be prime. Prove that if for some nonzero , then .
[Note: I’m not sure why D&F assume that is torsion. This proof doesn’t use that fact as far as I can tell. Read with caution.]
Recall that is an ideal (by this previous exercise). Since is a principal ideal domain, we have for some .
Consider the cyclic submodule generated by . Again, is an ideal, so for some . By our assumption, , so that for some . Since is prime, either or . If , then we have for some . Now , so that ; in particular, is a unit, and so . But then , a contradiction. So , and thus .
Since , we have , and so , as desired.