It is late at night and time for another silly question:
Is it true that a subset $S$ of an $R$-module $M$ is a submodule if and only if it is a subgroup of $M$ as an abelian group?
Of course, by definition a submodule is a subgroup, but I am wondering if the other direction is true as well. If the answer is no, are there specific types of $R$-modules in which this holds (other than simple groups please)?
No, it is not true.
For example, you should be able to find subgroups of $\mathbb Q$ which are not $\mathbb Q$-submodules.
Later. Suppose that $R$ is a ring such that every additive subgroup of $R$ is an $R$-submodule. Then the cyclic subgroup $S$ generated by $1$ is an $R$-submodule, and this implies easily that in fact $S=R$, that is, $R$ is generated as an abelian group by $1$. It follows that $R$ is isomorphic to $\mathbb Z/n\mathbb Z$ for some $n\in\mathbb N_0$. You can check that each of this rings has the desired property, so this gives you a complete list.