Projection on Cartesian Product of Modules

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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 $\struct {G_j, +_j, \circ_j}_R$.


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)