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 a unitary module, then so is $\mathcal L_R \left({G, H}\right)$.

Proof
From Group equals Center iff Abelian, the center of a commutative ring is the entire ring.

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