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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.

Thank you.

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    ... I see that as a kind of [Egg of Columbus](http://en.wikipedia.org/wiki/Egg_of_Columbus) phenomenon.2012-01-27

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