Correctness of the relation between free, torsion, torsion free, finitely generated module.

Question

Is the following relation and example correct?

(I modified the diagram on May 31, 2016.)

Answer 1

A free module is simply a module with a basis; it needn't be a finite basis. For instance, the countable direct sum of copies of $\Bbb Z$, $\bigoplus_{i=1}^\infty \Bbb Z$, is free with basis $e_i = (0,0, \dots, 1, \dots,)$, where the $1$ is in the $i$th position. In particular, free modules do not need to be finitely generated.

A similar example gives us an infinitely generated torsion ($\Bbb Z$-)module $\bigoplus_{n=2}^\infty \Bbb Z/n\Bbb Z$. This is torsion because, given an element $x=(a_2, \dots, a_k, 0,0,\dots)$, $k!x = 0$. You should prove that it isn't finitely generated.

It is correct that finitely generated torsion-free modules are free (over a PID), which is why "free" is in the intersection of those two bubbles; but it's important to know that free modules don't have to be finitely generated.