Image of Linear Transformation is Submodule
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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.
Let $\Img \phi$ denote the image set of $\phi$.
Then $\Img \phi$ is a submodule of $H$.
Proof
By Module is Submodule of Itself, $\struct {G, +_G, \circ_G}_R$ is a submodule of $\struct {G, +_G, \circ_G}_R$.
The result follows from Image of Submodule under Linear Transformation is Submodule.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Theorem $28.2$