faithful modules that aren't free

Question

What are some examples of faithful modules that are not free? It is clear that free modules are faithful but would the converse be true also?

Answer 1

It is well-known that if an ideal of a ring is free, then it is principal. Hence, every non-principal ideal of a domain is a faithful non-free module. For example, in the ring $R:=\mathbb{Z}[x, y]$, the ideals $\langle x+n, y+m\rangle$ for all $m, n\in\mathbb{Z}$ are faithful non-free $\mathbb{Z}$-modules. By this method we can construct many examples!!!

Answer 2

Counterexample:

$\mathbf{Q/Z}$ is a faithful torsion module.

Answer 3

$\prod_{n=2}^\infty \mathbb Z/n\mathbb Z$ is another faithful torsion $\mathbb Z$-module (hence certainly not free), apparently distinct from Bernard's since this one is uncountable.

For another one, try a nontrivial right ideal $T$ of $R=M_2(F_2)$ where $F_2$ is the field of two elements. $T$ is faithful since all nonzero unital modules of $R$ are faithful, and it can't be free purely because of cardinality. It also happens to be "torsion" in the sense that its elements all have nonzero annihilators.