Linear Transformation from Center of Scalar Ring

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

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

Let $Z \left({R}\right)$ be the center of the scalar ring $R$.

Let $\lambda \in Z \left({R}\right)$.

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

Proof
Let $\lambda \in \lambda \in Z \left({R}\right)$. Then:

Because $\lambda \in Z \left({R}\right)$, $\lambda$ commutes with all elements of $R$.

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

Thus: