Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings

Theorem
Let $$R$$ be a commutative ring.

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

Let $$\mathcal {L}_R \left({G, H}\right)$$ be the set of all linear transformations from $$G$$ to $$H$$.

Then $$\mathcal {L}_R \left({G, H}\right)$$ is a submodule of the $R$-module $H^G$.

If $$H$$ is unitary, then so is $$\mathcal {L}_R \left({G, H}\right)$$.

Proof
From Center of Abelian Group is Whole Group, the center of a commutative ring is (obviously) the entire ring.

The result follows from Linear Transformation from Center of Scalar Ring.