Homomorphic Image of R-Module is R-Module

Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G, \circ_G}_R$ be an $R$-module.

Let $\struct {H, +_H, \circ_H}_R$ be an $R$-algebraic structure.

Let $\phi: G \to H$ be a homomorphism.

Then the homomorphic image of $\phi$ is an $R$-module.

Proof
Let us write $\phi \sqbrk G$ to denote the homomorphic image of $\phi$.

From Image of Group Homomorphism is Subgroup, $\phi \sqbrk G$ is a subgroup of $\struct {H, +_H}$.

For any $\map \phi g$ and $\map \phi {g'}$ in $\phi \sqbrk G$, we have:

hence $\phi \sqbrk G$ is an abelian group.

Now we can turn to showing that $\phi \sqbrk G$ is an $R$-module.

To do this, we take the $R$-module axioms in turn.

It is to be shown that for all $\lambda \in R$ and $\map \phi g, \map \phi {g'} \in \phi \sqbrk G$:


 * $\lambda \circ_H \paren {\map \phi g +_H \map \phi {g'} } = \paren {\lambda \circ_H \map \phi g} +_H \paren {\lambda \circ_H \map \phi {g'} }$

Compute, using that $\phi$ is a homomorphism repetitively:

It is to be shown that for all $\lambda, \mu \in R$ and $\map \phi g \in \phi \sqbrk G$:


 * $\paren {\lambda +_R \mu} \circ_H \map \phi g = \paren {\lambda \circ_H \map \phi g} +_H \paren {\mu \circ_H \map \phi g}$

Compute, using that $\phi$ is a homomorphism repetitively:

It is to be shown that for all $\lambda, \mu \in R$ and $\map \phi g \in \phi \sqbrk G$:


 * $\paren {\lambda \times_R \mu} \circ_H \map \phi g = \lambda \circ_H \paren {\mu \circ_H \map \phi g}$

Compute, using that $\phi$ is a homomorphism repetitively:

Having verified that $\phi \sqbrk G$ satisfies the three axioms, we conclude it is an $R$-module.