If $M$ is an $R-$module, and id $N$ is any subset of $M$ satisfying $rn \in N$ for all $r\in R$ & $ n\in N$ , then does that imply $N$ is a subgroup of $M$?
I tried to prove the closure part but couldn't complete it. I didn't have the sufficient information to do. So I think it wouldn't be a subgroup. But I am unable to find any counter-example . Please give me a counterexample, or show that it is a subgroup.
Edit-I think I got a counterexample. Consider $\mathbb{Z}$ as a module over $\mathbb{Z}$ and consider the set $ N=\{0,\pm 4, \pm 6,\pm 8,\pm10\dots \dots\}$
Your counterexample, if I understand well what it consists of (all even integers, except $\pm 2$), works.
More generally, the assertion is false because it would imply that the union of two subvectorspaces or two submodules is a sub vectorspace/ submodules. It is well-known that, for instance, the union of two subgroups is not a subgroup, unless one of them is contained in the other.