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

Theorem

Let $R$ be a commutative ring with unity whose unity is $1_R$.

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 $H$.

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

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

Proof

From Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings, $\map {\LL_R} {G, H}$ is a module.

It remains to be shown that $\map {\LL_R} {G, H}$ is a unitary module, that is:

$\forall \phi \in \map {\LL_R} {G, H}: 1_R \circ \phi = \phi$

So, let $\struct {H, +_H, \circ}_R$ be a unitary $R$-module.

Then, by Unitary Module Axiom $\text {UM} 4$: Unity of Scalar Ring:

$\forall x \in H: 1_R \circ x = x$

Thus:

\(\ds \map {\paren {1_R \circ \phi} } x\)

\(=\)

\(\ds 1_R \circ \paren {\map \phi x}\)

Definition of Composition of Mappings

\(\ds \)

\(=\)

\(\ds \map \phi x\)

Unitary Module Axiom $\text {UM} 4$: Unity of Scalar Ring

Hence the result.

$\blacksquare$

Sources

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