Every $R$-module is free $\implies$ $R$ is a division ring
Prove that if a (generally noncommutattive) ring $R$, any $R$-module is free then $R$ is a field.
The commutative case is fairly easy, but I don't know how to deal with the noncommutative one. What could be the tools, or in what context is this problem solved most clearly?
Although I am somehow a novice in noncommutative ring theory, the problem looks very interesting and I am willing to study something new even for this problem only.