## Every finite abelian group is a torsion ZZ-module

Let $R$ be a ring. A left $R$-module $M$ is called torsion if $\mathsf{Tor}_R(M) = M$; that is, if for all $m \in M$, there exists a nonzero $r \in R$ such that $r \cdot m = 0$. Prove that every finite abelian group is torsion. Exhibit an infinite abelian group that is torsion.

Suppose $M$ is a finite abelian group (written additively) of order $k$. Make $M$ into a left $\mathbb{Z}$-module in the usual way- $r \cdot m = rm$. Note that $k \cdot m = km = 0$ for all $m \in M$ by Lagrange’s Theorem. Thus $M = \mathsf{Tor}_R(M)$, and so $M$ is torsion.

Consider now the ring $M = \prod_{\mathbb{N}} \mathbb{Z}/(2)$. As an abelian group, $M$ is a (left) $\mathbb{Z}$-module. Moreover, $2 \cdot (a_i) = (2 \cdot a_i) = (0)$ for all $(a_i)$; hence $M$ is torsion.