Projection onto Ideal of External Direct Sum of Rings

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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 $\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$ be their 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 Sum of Rings we already have that $R'_k$ is an ideal of $R$.

The result follows by application of Projection is Epimorphism.

$\blacksquare$


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