Exhibit an integral domain and a nonzero torsion module over such that . Prove that if is a finitely generated torsion -module ( a domain), then .
First consider the direct sum as a -module in the usual way. Every element of is annihilated by , where is the largest nonzero component of . However, for all , there exist elements not annihilated by . For example, if , then the element with in the component an 0 elsewhere is not annihilated by .
Suppose now that is a finitely generated torsion -module; say , with . For each , there exists a nonzero element such that , since is torsion. Let . Since is an integral domain, , and certainly . Thus .