Projection onto Ideal of External Direct Sum of Rings

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

Let $$\left({R, +, \circ}\right) = \prod_{k=1}^n \left({R_k, +_k, \circ_k}\right)$$ be their external direct product.

For each $$k \in \left[{1 \,. \, . \, n}\right]$$, let:


 * $$R'_k = \left\{{\left({x_1, \ldots, x_n}\right) \in R: \forall j \ne k: x_j = 0}\right\}$$

Let $$\operatorname{pr}_k: R \to R'_k$$ be the projection on the $k$th coordinate of $$\left({R, +, \circ}\right)$$ onto $$R'_k$$.

Then $$\operatorname{pr}_k$$ is an epimorphism.

Proof
From Ideal of External Direct Product of Rings we already have that $$R'_k$$ is an ideal of $$R$$.

The result follows by application of Projections are Epimorphisms.