Canonical Injection from 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 $$\operatorname{in}_k: R_k \to R$$ be the canonical injection on the $k$th coordinate from $$R_k$$ into $$\left({R, +, \circ}\right)$$.

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:
 * $$\operatorname{in}_k: R_k \to R'_k$$ is an isomorphism;
 * Its inverse is the restriction of $$\operatorname{pr}_k$$ to $$R'_k$$

Proof
From Ideal of External Direct Product of Rings we have that $$R'_k$$ is an ideal of $$R$$, and thus a subring of $$R$$.

The result follows by application of Canonical Injections are Monomorphisms.