Linear Transformation of 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.
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 $G$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Theorem $28.2$