Let $A$ be an algebra over a field $K$ of finite dimension. Let $M$ be a finitely generated $A$-module. My question is about the ring ${\rm End}_A(M)$ of $A$-endomorphisms of $M$. I want to show that it is a $K$-algebra.
To show that $K$ lies in the center of ${\rm End}_A(M)$, I look at the ring homomorphism $K\to {\rm End}_A(M): \alpha\mapsto \alpha 1_M$, where $1_M$ is the identity mapping on $M$. I can show that $\alpha1_M$ commutes with every $f\in {\rm End}_A(M)$. But is this map 1-1? How can I show that if $\alpha1_M=0$ then $\alpha=0$?
Hint: if $f\colon K\to R$ is a ring homomorphism (mapping $1$ to $1$) and $R$ is not a zero ring, then $f$ is one-to-one whenever $K$ is a field.