Linear Transformation from Center of Scalar Ring

Theorem
Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules.

Let $\phi: G \to H$ be a linear transformation.

Let $\map Z R$ be the center of the scalar ring $R$.

Let $\lambda \in \map Z R$.

Then $\lambda \circ \phi$ is a linear transformation.

Proof
Let $\lambda \in \lambda \in \map Z R$.

Then:

Because $\lambda \in \map Z R$, $\lambda$ commutes with all elements of $R$.

So $\forall \mu \in R: \lambda \circ \mu = \mu \circ \lambda$.

Thus: