# 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.

### Image of Submodule under Linear Transformation is Submodule

Let $M$ be a submodule of $G$.

Then $\phi \sqbrk M$ is a submodule of $H$.

### Preimage of Submodule under Linear Transformation is Submodule

Let $N$ be a submodule of $H$.

Then $\phi^{-1} \sqbrk N$ is a submodule of $G$.

### Image of Linear Transformation is Submodule

Let $\Img \phi$ denote the image set of $\phi$.

Then $\Img \phi$ is a submodule of $H$.

### Kernel of Linear Transformation is Submodule

Let $\map \ker \phi$ denote the kernel of $\phi$.

Then $\map \ker \phi$ is a submodule of $G$.