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!

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    $\begingroup$ A big discussion of vector spaces over division rings is here: math.stackexchange.com/questions/45056/… $\endgroup$ – ncmathsadist Apr 27 '15 at 18:05
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    $\begingroup$ Did not see that discussion; thanks! $\endgroup$ – user143137 Apr 27 '15 at 18:12
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    $\begingroup$ Have you tried going through your favorite proof that every vector space has a basis, and checking whether it will work for arbitrary division rings? If it does, then you've answered your question. If it doesn't, then pinpointing the step that fails will give you a crisper target for constructing a counterexample. $\endgroup$ – Henning Makholm Apr 27 '15 at 18:14

Not only it holds true, but it is a characterisation of division rings(unfortunately you need Zorn's Lemma).


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