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 $\displaystyle \left({R, +, \circ}\right) = \prod_{k=1}^n \left({R_k, +_k, \circ_k}\right)$ 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.