Let be an integral domain, let be a (left, unital) -module, and let be a submodule. Prove that .
Suppose is a cardinality-maximal linearly independent set, and choose such that is a cardinality-maximal linearly independent set and and have the same cardinality. Note in particular that is empty. We claim that is linearly independent in . To see this, suppose . Mod , we see that , so that . But then , so that . Thus is linearly independent in . In particular, .
Now let be the submodule generated by . We claim that is torsion. Suppose to the contrary that is not torsion; that is, that there exists such that if , then . In other words, if , then , and hence . Thus is linearly independent. If , then we violate the maximality of , and if , then we violate the maximality of . So must be torsion. By this previous exercise, .