Characterization of finitely generated $\mathbb{Z}$-modules with the property that each submodule is a direct summand

Question

I want to characterize all finitely generated $\mathbb{Z}$-modules $M$ with the property that each submodule of $M$ is a direct summand of $M$. I think the module has to be torsion but I couldn't say more. Any help would be great. Thanks.

Answer 1

Use the primary decomposition of finitely generated abelian groups. Since $\mathbb Z$ has subgroups which are not direct summands, you can restrict to the torsion groups. The groups of the form $\mathbb Z/p^m\mathbb Z$ with $m\ge 2$ have subgroups which are not direct summands, so your abelian group is isomorphic to $\mathbb Z/p_1\mathbb Z\times\cdots\times\mathbb Z/p_r\mathbb Z$ with $p_i$ distinct primes, that is, isomorphic to $\mathbb Z/p_1\cdots p_r\mathbb Z$ with $p_i$ distinct primes. (You can check that these abelian groups satisfies the required condition.)