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
By definition of linear transformation, we need to show that:
 * $(1): \quad \forall x, y \in G: \map {\paren {\lambda \circ \phi} } {x +_G y} = \lambda \circ \map \phi x +_H \lambda \circ \map \phi y$
 * $(2): \quad \forall x \in G: \forall \mu \in R: \map {\paren {\lambda \circ \phi} } {\mu \circ x} = \mu \circ \map {\paren {\lambda \circ \phi} } x$

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

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: