I am currently trying to answer the following true/false question:
True or False: Every module over a division ring $R$ is free.
I know the result would be true if $R$ is a field (ie a commutative division ring), but I'm unsure if the statement is necessarily true for non-commutative division rings. I'm guessing the best way to try and find a counterexample is to let $R = \mathbb{H}$ (real quaternions), but I don't really have any ideas / experience with examples of $\mathbb{H}$-modules.
So is this statement actually true, or is their a example (preferably of a $\mathbb{H}$-module) which is not free?
Many thanks!
Best Answer
Not only it holds true, but it is a characterisation of division rings(unfortunately you need Zorn's Lemma).