Projection on Cartesian Product of Modules
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Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {G, +, \circ}_R$ be the cartesian product of a sequence $\sequence {\struct {G_n, +_n, \circ_n}_R}$ of $R$-modules.
Then for each $j \in \closedint 1 n$, the projection $\pr_j$ on the $j$th co-ordinate is an epimorphism from $\struct {G, +, \circ}_R$ onto $\struct {G_j, +_j, \circ_j}_R$.
Proof
To demonstrate that $\pr_j$ is an epimorphism, we need to show that:
- $(1): \quad \pr_j$ is a surjection
- $(2): \quad \forall x, y \in G: \map {\pr_j} {x + y} = \map {\pr_j} x +_j \map {\pr_j} y$
- $(3): \quad \forall x \in G: \forall \lambda \in R: \map {\pr_j} {\lambda \circ x} = \lambda \circ_j \map {\pr_j} x$
Criteria $(1)$ and $(2)$ are a direct application of Projection is Epimorphism.
Let $x = \tuple {x_1, x_2, \ldots, x_j, \ldots, x_n}$
Then we have:
\(\ds \forall x \in G: \forall \lambda \in R: \, \) | \(\ds \map {\pr_j} {\lambda \circ x}\) | \(=\) | \(\ds \map {\pr_j} {\lambda \circ \tuple {x_1, x_2, \ldots, x_j, \ldots, x_n} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\pr_j} {\tuple {\lambda \circ_1 x_1, \lambda \circ_2 x_2, \ldots, \lambda \circ_j x_j, \ldots, \lambda \circ_n x_n} }\) | Definition of Cartesian Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \circ_j x_j\) | Definition of Projection (Mapping Theory) | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \circ_j \map {\pr_j} x\) | Definition of Projection (Mapping Theory) |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.7$