Linear Transformation from Center of Scalar Ring

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:

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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:

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