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

Theorem
Let $R$ be a commutative ring.

Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules.

Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$.

Then $\map {\LL_R} {G, H}$ is a submodule of the $R$-module $H^G$.

If $H$ is a unitary module, then so is $\map {\LL_R} {G, H}$.

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

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