Projection onto Ideal of External Direct Sum of Rings

Theorem
Let $\struct {R_1, +_1, \circ_1}, \struct {R_2, +_2, \circ_2}, \ldots, \struct {R_n, +_n, \circ_n}$ be rings.

Let $\ds \struct {R, +, \circ} = \prod_{k \mathop = 1}^n \struct {R_k, +_k, \circ_k}$ be their direct product.

For each $k \in \closedint 1 n$, let:


 * ${R_k}' = \set {\tuple {x_1, \ldots, x_n} \in R: \forall j \ne k: x_j = 0}$

Let $\pr_k: R \to {R_k}'$ be the projection on the $k$th coordinate of $\struct {R, +, \circ}$ onto ${R_k}'$.

Then $\pr_k$ is an epimorphism.

Proof
From Ideal of External Direct Sum of Rings we already have that ${R_k}'$ is an ideal of $R$.

The result follows by application of Projection is Epimorphism.

Also see

 * Canonical Injection from Ideal of External Direct Sum of Rings