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:

$$ $$ $$ $$