Module of All Mappings is Module
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Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {G, +_G, \circ}_R$ be an $R$-module.
Let $S$ be a set.
Let $\struct {G^S, +_G', \circ}_R$ be the module of all mappings from $S$ to $G$.
Then $\struct {G^S, +_G', \circ}_R$ is an $R$-module.
Proof
To show that $\struct {G^S, +_G', \circ}_R$ is an $R$-module, we verify the following:
$\forall f, g, \in G^S, \forall \lambda, \mu \in R$:
- $(1): \quad \lambda \circ \paren {f +_G' g} = \paren {\lambda \circ f} +_G' \paren {\lambda \circ g}$
- $(2): \quad \paren {\lambda +_R \mu} \circ f = \paren {\lambda \circ f} +_G \paren {\mu \circ f}$
- $(3): \quad \paren {\lambda \times_R \mu} \circ f = \lambda \circ \paren {\mu \circ f}$
Criterion 1
- $(1): \quad \lambda \circ \paren {f +_G' g} = \paren {\lambda \circ f} +_G' \paren {\lambda \circ g}$
Let $x \in S$.
Then:
\(\ds \lambda \circ \paren {\map {\paren {f +_G' g} } x}\) | \(=\) | \(\ds \lambda \circ \paren {\map f x +_G \map g x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\lambda \circ \map f x} +_G \paren {\lambda \circ \map g x}\) | Definition of Module | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map {\paren {\lambda \circ f} } x} +_G \paren {\map {\paren {\lambda \circ g} } x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\paren {\lambda \circ f} +_G' \paren {\lambda \circ g} } } x\) |
Thus $(1)$ holds.
$\Box$
Criterion 2
- $(2): \quad \paren {\lambda +_R \mu} \circ f = \paren {\lambda \circ f} +_G \paren {\mu \circ f}$
Let $x \in S$.
\(\ds \map {\paren {\paren {\lambda +_R \mu} \circ f} } x\) | \(=\) | \(\ds \paren {\lambda +_R \mu} \circ \paren {\map f x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\lambda \circ \map f x} +_G \paren {\mu \circ \map f x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\lambda \circ f} } x +_G \map {\paren {\mu \circ f} } x\) |
Thus $(2)$ holds.
$\Box$
Criterion 3
- $(3): \quad \paren {\lambda \times_R \mu} \circ f = \lambda \circ \paren {\mu \circ f}$
\(\ds \map {\paren {\paren {\lambda \times_R \mu} \circ f} } x\) | \(=\) | \(\ds \paren {\lambda \times_R \mu} \circ \paren {\map f x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \circ \paren {\mu \circ \map f x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \circ \paren {\map {\paren {\mu \circ f} } x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\lambda \circ \paren {\mu \circ f} } } x\) |
Thus $(3)$ holds.
$\Box$
Hence the result.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.4$