Module endomorphisms

Question

Let $R$ be a ring, $M_R$ a right module over $R$.

Let $S:=$ End($M_R)$

I have shown that $R \subseteq$ End$(_SM)$

(pick $r \in R$, then define $f_r(m)$ via $m \mapsto mr$ show $f_r \in$ End$(_SM$))

I have to show proper containment via example.

So an endomorphism of $_SM$ (so scalars are also Endomorphisms) that is not a ring element, I get stuck here, so I need to find an explicit map that is an endomorphism but cannot be a ring element?

Answer 1

First, the homomorphism $R\to \mathrm{End}({}_SM)$ is an embedding only if $M_R$ is faithful, that is $Mr=0\implies r=0$.

For an example, let $R=\Bbb Z,\ M=\Bbb Q$, then $S\cong\Bbb Q$ as a $\Bbb Z$-endomorphism of $\Bbb Q$ is uniquely determined by the image of $1$.
But we also have $\mathrm{End}({}_SM)\ =\ \mathrm{End}({}_{\Bbb Q}\Bbb Q)\ \cong\ \Bbb Q$ which is a proper superset of $R=\Bbb Z$, and a specific map you're looking for is multiplying by any noninteger rational number.