Linear Transformation of Submodule

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

Let $\struct {G, +_G, \circ_G}_R$ and $\struct {H, +_H, \circ_H}_R$ 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
Let $e_H$ be the identity of $\struct {H, +_H}$.

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 ...