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.

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