# 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 $G$, $\phi \sqbrk M$ is a submodule of $H$
$(2): \quad$ If $N$ is a submodule of $H$, $\phi^{-1} \sqbrk N$ 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 Codomain Preserves Identity, $\map \phi {e_G} = e_H$
$(2): \quad$ By Homomorphism with Identity Preserves Inverses, $\map \phi {-x} = -\map \phi x$.

From Epimorphism preserves Modules and definition of surjection, it follows that as $M$ is a submodule of $G$, then $\phi \sqbrk M$ is a submodule of $H$.

The result follows ...