Linear_Transformation_of_Submodule

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 $G$.