Let $N$ a $R$ submodule of $M$, then $\operatorname{End}_{R}(N)$ is subring of $\operatorname{End}_{R}(M)$. Basically I just need to do is the subring test:
https://proofwiki.org/wiki/Subring_Test
where (1), (2) and (3) seems easy to prove. But Im doubting if it always happens that $\operatorname{End}_{R}(N) \subset\operatorname{End}_{R}(M)$ as if I take some $f \in \operatorname{End}_{R}(N) $ how I can extend this to $f \in \operatorname{End}_{R}(M) $?? So is it true that $\operatorname{End}_{R}(N)$ is subring of $\operatorname{End}_{R}(M)$ for my hypothesis or there is a counterexample? If this is not true does anyone know something about necessity conditions? Thanks!
Obviously not.
Consider $R=M=\mathbb Z/4\mathbb Z$ and $N=2\mathbb Z/4\mathbb Z$.
Then $End(N_R)$ is the field of two elements, and has characteristic $2$, and $End(M_R)\cong \mathbb Z/ 4\mathbb Z$ has characteristic $4$, so the former can't be a subring of the latter.
Another natural thing to investigate is if $N_R$ is a fully invariant submodule of $M_R$ (meaning $f:N\subseteq N$ for every $f\in End(M_R)$. In that case, every endomorphism of $M$ restricts to an endomorphism of $N$, and you have a map from $End(M_R)\to End(N_R)$, of some sort, which I will let you investigate.