Finite Direct Product of Modules is Module/Proof 2
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Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {G_1, +_1, \circ_1}_R, \struct {G_2, +_2, \circ_2}_R, \ldots, \struct {G_n, +_n, \circ_n}_R$ be $R$-modules.
Let:
- $\ds G = \prod_{k \mathop = 1}^n G_k$
be their direct product.
Then $G$ is a module.
Proof
Module Axiom $\text M 1$: Distributivity over Module Addition
Let $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in G$.
| \(\ds \lambda \circ \paren {x + y}\) | \(=\) | \(\ds \lambda \circ \paren {\tuple {x_1, x_2, \ldots, x_n} + \tuple {y_1, y_2, \ldots, y_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \circ \tuple {x_1 +_1 y_1, x_2 +_2 y_2, \ldots, x_n +_n y_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\lambda \circ_1 \paren {x_1 + y_1}, \lambda \circ_2 \paren {x_2 + y_2}, \ldots, \lambda \circ_n \paren {x_n + y_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\lambda \circ_1 x_1, \lambda \circ_2 x_2, \ldots, \lambda \circ_n x_n} + \tuple {\lambda \circ_1 y_1, \lambda \circ_2 y_2, \ldots, \lambda \circ_n y_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\lambda \circ \tuple {x_1, x_2, \ldots, x_n} } + \paren {\lambda \circ \tuple {y_1, y_2, \ldots, y_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\lambda \circ x} + \paren {\lambda \circ y}\) |
So $(1)$ holds.
$\Box$
Module Axiom $\text M 2$: Distributivity over Scalar Addition
Let $x = \tuple {x_1, x_2, \ldots, x_n} \in G$.
| \(\ds \paren {\lambda +_R \mu} \circ x\) | \(=\) | \(\ds \paren {\lambda +_R \mu} \circ \tuple {x_1, x_2, \ldots, x_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\paren {\lambda +_R \mu} \circ_1 x_1, \paren {\lambda +_R \mu} \circ_2 x_2, \ldots, \paren {\lambda +_R \mu} \circ_n x_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\paren {\lambda \circ_1 x_1} +_1 \paren {\mu \circ_1 x_1}, \paren {\lambda \circ_2 x_2} +_2 \paren {\mu \circ_2 x_2}, \ldots, \paren {\lambda \circ_n x_n} +_n \paren {\mu \circ_n x_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\lambda \circ_1 x_1, \lambda \circ_2 x_2, \lambda \circ_n x_n} + \tuple {\mu \circ_1 x_1, \mu \circ_2 x_2, \ldots, \mu \circ_n x_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\lambda \circ_1 \tuple {x_1, x_2, \ldots, x_n} } + \paren {\mu \circ_1 \tuple {x_1, x_2, \ldots, x_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\lambda \circ x} + \paren {\mu \circ x}\) |
So $(2)$ holds.
$\Box$
Module Axiom $\text M 3$: Associativity
Let $x = \tuple {x_1, x_2, \ldots, x_n} \in G$.
| \(\ds \paren {\lambda \times_R \mu} \circ x\) | \(=\) | \(\ds \paren {\lambda \times_R \mu} \circ \tuple {x_1, x_2, \ldots, x_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\paren {\lambda \times_R \mu} \circ_1 x_1, \paren {\lambda \times_R \mu} \circ_2 x_2, \ldots, \paren {\lambda \times_R \mu} \circ_n x_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\lambda \circ_1 \paren {\mu \circ_1 x_1}, \lambda \circ_2 \paren {\mu \circ_2 x_2}, \ldots, \lambda \circ_n \paren {\mu \circ_n x_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \circ \tuple {\mu \circ_1 x_1, \mu \circ_2 x_2, \ldots, \mu \circ_n x_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \circ \paren {\mu \circ \tuple {x_1, x_2, \ldots, x_n} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lambda \circ \paren {\mu \circ x}\) |
So $(3)$ holds.
$\Box$
Hence the result.
$\blacksquare$