Linear Transformation of Submodule

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

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

Then:
 * 1) If $$M$$ is a submodule of $$G$$, $$\phi \left({M}\right)$$ is a submodule of $$H$$;
 * 2) If $$N$$ is a submodule of $$H$$, $$\phi^{-1} \left({N}\right)$$ is a submodule of $$G$$;
 * 3) The range of $$\phi$$ is a submodule of $$H$$;
 * 4) 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) By Homomorphism with Cancellable Range Preserves Identity, $$\phi \left({e_G}\right) = e_H$$;
 * 2) By Homomorphism with Identity Preserves Inverses, $$\phi \left({-x}\right) = -\phi \left({x}\right)$$.


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

The result follows ...