Canonical Injection from Ideal of External Direct Sum of Rings

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Theorem

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

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


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

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


Let $\inj_k: R_k \to R$ be the canonical injection on the $k$th coordinate from $R_k$ into $\struct {R, +, \circ}$.

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


Then:

$\inj_k: R_k \to R'_k$ is an isomorphism
Its inverse is the restriction of $\pr_k$ to $R'_k$.


Proof

From Ideal of External Direct Sum 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 Injection is Monomorphism.

$\blacksquare$


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