Let be a ring. A left -module is called torsion if ; that is, if for all , there exists a nonzero such that . Prove that every finite abelian group is torsion. Exhibit an infinite abelian group that is torsion.
Suppose is a finite abelian group (written additively) of order . Make into a left -module in the usual way- . Note that for all by Lagrange’s Theorem. Thus , and so is torsion.
Consider now the ring . As an abelian group, is a (left) -module. Moreover, for all ; hence is torsion.