$A$-linear maps between $A$-modules where $A$ is a $K$-algebra and $K$ is a commutative ring

Question

This is from Alexander Zimmermann's Represenation Theory. How can we talk about "$K$-linearity" of an $A$-module homomorphism $\alpha:M\to N$?

Answer 1

Note first that $M$ is a $K$-module by the action

$$k.m:= (k1_A)m$$

Similarly $N$ is a $K$-module.

Let $\alpha: M \to N$ be an $A$-module homomorphism. Let $k \in K, m \in M$

$$\alpha(k.m)=\alpha((k1_A)m) = (k1_A)\alpha(m) = k. \alpha(m)$$

so $\alpha$ is $K$-linear.

In short, this comes down to saying that $A$ is a $K$-algebra implies $K \subseteq Z(A) \subseteq A$ so a map that is $A$-linear is automatically also $K$-linear.