Let $M $ be a left module over a ring $R $ with idenitty. If $R $ is commutative, we know that for every $r\in R $ ,$End_{R}(M)$ contains the function that sends $m $ to $rm $.
If $R $ is not commutative, then the only homomorphisms inside $End_{R}(M)$ that I can think of are the zero and identity maps.
Are there other natural examples of maps in $End_{R}(M)$ thatI m not aware of ?
Is it possible that for a nontrivial pair of $M,R $ we have that $End_{R}(M)=\{0,1_{M}\}$
Thank you
Ok after Pedro s comment, I realize now that the endomorphism ring will also contain the maps that send $m $ to $m+m $ and that send $m$ to $m+m+m $ and ........ I m still hoping to see more interesting maps
Let $F = \mathbb F_p$ for $p$ a prime number, and let $$R = \begin{pmatrix}* & * & \cdots & * \\ 0 & * & \cdots & * \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & *\end{pmatrix}$$ be the ring of upper triangular matrices with entries in $F$ (of some fixed size). $R$ acts on $$M = \begin{pmatrix} * \\ 0 \\ \vdots \\ 0\end{pmatrix}$$ in the natural way. We have $\operatorname{End}_R(M) = F$. Since $\mathbb Z \twoheadrightarrow F$, it follows that the endomorphisms of $M$ consist only of those of the form $m \mapsto m+ m + \cdots + m$.
Does that meet your criterion of "nontrivial"?