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

Theorem

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 $H^G$.

Let $\struct {H, +_H, \circ}_R$ be a unitary module.

Then $\map {\LL_R} {G, H}$ is also a unitary module.

By definition of abelian group, the center of a commutative ring $R$ is $R$ itself.

So, we need to show that every $\phi \in \map {\LL_R} {G, H}$ fulfils the conditions to be a linear transformation:

$(1): \quad \forall x, y \in G: \map {\paren {\lambda \circ \phi} } {x +_G y} = \lambda \circ \map \phi x +_H \lambda \circ \map \phi y$

$(2): \quad \forall x \in G: \forall \mu \in R: \map {\paren {\lambda \circ \phi} } {\mu \circ x} = \mu \circ \map {\paren {\lambda \circ \phi} } x$

for all $\phi \in \map {\LL_R} {G, H}$.

Hence the result.

$\blacksquare$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations