Let be an integral domain and let be a finitely generated torsion left -module. Prove that .
Since is finitely generated, we have for some finite set . Since is torsion, for each there exists nonzero such that . Since is an integral domain, is nonzero. Now if , . Thus , and in particular .
Now consider the abelian group (hence left -module) . Suppose there exists a nonzero element . Choose such that does not divide . Now , so that is an integer. But then , and thus , so that divides , a contradiction. So . (So we can conclude that is not finitely generated as an abelian group, a result we proved previously with a bit more effort.)