I am trying to do the following exercise:
Prove that if $A$ is a $k-$algebra and $M$ is a module then the product $\lambda * m=\tau(\lambda)m$ ($\tau$ is the morphism from $K$ to $Z(A)$) defines on $M$ a $k-$vector space structure. Moreover, the image of the canonical morphism $\rho:A \to End_{ \mathbb Z}(M)$ is contained in the subring $End_k(M) \subset End_{ \mathbb Z}(M)$.
I could show that $M$ is a $k-$module (a $k-$vector space), however I am a bit confused with the "canonical morphism" $\rho:A \to End_{ \mathbb Z}(M)$ and the rings $End_k(M)$ and $End_{ \mathbb Z}(M)$. Since $M$ is an $A-$module, we have a ring morphism $\beta:A \to End(M)$ but what it means the notation $End_{ \mathbb Z}(M)$? I have the same confusion with the notation $End_k(M)$, I have proved that $M$ is a $K-$ module, this means there is a ring morphism $\phi:K \to End(M)$, but what do they mean by $End_k(M)$?.
I would appreciate if someone could clear this up for me. Thanks in advance.
$End_Z(M)=\{f \colon M \to M|f(a+b)=f(a)+f(b)\ \forall\ a,b \in M\}$
$End_k(M)=\{f \in End_Z(M)|f(ka)=kf(a) \forall\ a \in M,\forall k \in K\}$