Finite Direct Product of Unitary Modules is Unitary Module
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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 unitary $R$-modules.
Let:
- $\ds G = \prod_{k \mathop = 1}^n G_k$
be their direct product.
Then $G$ is a unitary module.
Proof 1
This is a special case of Direct Product of Unitary Modules is Unitary Module.
Proof 2
From Finite Direct Product of Modules is Module we have that $G$ is a module.
It remains to be shown that:
- $\forall x \in G: 1_R \circ x = x$
Let $x = \tuple {x_1, x_2, \ldots, x_n} \in G$.
Then:
\(\ds 1_R \circ x\) | \(=\) | \(\ds 1_R \circ \tuple {x_1, x_2, \ldots, x_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {1_R \circ x_1, 1_R \circ x_2, \ldots, 1_R \circ x_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1, x_2, \ldots, x_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x\) |
Hence the result.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.5$