Linear Transformation of Submodule

Theorem
Let $G$ and $H$ be $R$-modules.

Let $\phi: G \to H$ be a linear transformation.

Then:
 * $(1): \quad$ If $M$ is a submodule of $G$, $\phi \left({M}\right)$ is a submodule of $H$
 * $(2): \quad$ If $N$ is a submodule of $H$, $\phi^{-1} \left({N}\right)$ is a submodule of $G$
 * $(3): \quad$ The codomain of $\phi$ is a submodule of $H$
 * $(4): \quad$ The kernel of $\phi$ is a submodule of $G$.

Proof
Since a linear transformation $\phi: G \to H$ is, in particular, a homomorphism from the group $G$ to the group $H$, it follows that:


 * $(1): \quad$ By Homomorphism with Cancellable Range Preserves Identity, $\phi \left({e_G}\right) = e_H$
 * $(2): \quad$ By Homomorphism with Identity Preserves Inverses, $\phi \left({-x}\right) = -\phi \left({x}\right)$.


 * From Morphisms from Modules and Surjection iff Image equals Codomain, it follows that as $M$ is a submodule of $G$, then $\phi \left({M}\right)$ is a submodule of $H$.

The result follows ...