Characterization of a vector space over an associative division ring Let $M$ be a (left) module over an associative division ring $R$.
Then it has the following properties.
1) For every submodule $N$ of $M$, there exists a submodule $L$ such that $M = N + L$
and $M \cap L = 0$.
2) Every finitely generated submodule has a composition series.
Now let $M \neq 0$ be a (left) faithful module over an associative ring $R$ with unity 1.
Suppose $M$ satisfies the above conditions.
Is $R$ necessarily a division ring?
EDIT Related question:Module over a ring which satisfies Whitehead's axioms of projective geometry

Let $R=\mathbb{Z}$ so that $R$-modules are abelian groups and submodules are subgroups.  Let $M$ be a cyclic group of prime order.  Then $M$ is simple, so 1) is satisfied, and every finite group has a composition series, so 2) is satisfied.  However, $R$ is not a division ring.
