Problem: Let $A$ be a unitary module over a division ring $R$. Prove that for each submodule $B$, there exist a submodule $C$ such that $A=B\oplus C$.
Anyone can help me in this problem? I really don't see the connection between the premises and the conclusion. I don't even know where to start, I mean I can start with the premises but it is just until there I don't know where to proceed. Any help is much appreciated.
Note that every unitary module A over a division ring R has a basis (and therefore a free R-module).Existence of basis implies A is the internal direct sum of cyclic submodules based on the equivalent conditions in the definition of free module. If you can show that every submodule B of A is cyclic and is among the cyclic submodule mentioned above then I think that proves it. Our choice of C will be the internal direct sum of the remaining cyclic submodules. This is just my take on the matter, I haven't fully researched on this yet. Hope this helps.