Canonical Injection into Cartesian Product of Modules/Proof 1

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 canonical injection $\inj_j$ from $\struct {G_j, +_j, \circ_j}_R$ into $\struct {G, +, \circ}_R$ is a monomorphism.

Proof

To demonstrate that $\inj_j$ is an epimorphism, we need to show that:

$(1): \quad \inj_j$ is an injection

$(2): \quad \forall x, y \in G_j: \map {\inj_j} {x +_j y} = \map {\inj_j} x + \map {\inj_j} y$

$(3): \quad \forall x_j \in G_j: \forall \lambda \in R: \map {\inj_j} {\lambda \circ_j x_j} = \lambda \circ \map {\inj_j} {x_j}$

Criteria $(1)$ and $(2)$ are a direct application of Canonical Injection is Monomorphism.

Let $x_j \in G_j$ be arbitrary.

Then we have:

\(\ds \forall x \in G_j: \forall \lambda \in R: \, \)

\(\ds \map {\inj_j} {\lambda \circ_j x_j}\)

\(=\)

\(\ds \tuple {e_1, e_2, \ldots, \lambda \circ_j x_j, \ldots, e_n}\)

Definition of Canonical Injection

\(\ds \)

\(=\)

\(\ds \lambda \circ \tuple {e_1, e_2, \ldots, x_j, \ldots, e_n}\)

Definition of Cartesian Product

\(\ds \)

\(=\)

\(\ds \lambda \circ \map {\inj_j} {x_j}\)

Definition of Canonical Injection

$\blacksquare$