External Direct Product of Ringoids is Ringoid

Theorem
Let $\struct {R_1, +_1, \circ_1}, \struct {R_2, +_2, \circ_2}, \ldots, \struct {R_n, +_n, \circ_n}$ be ringoids.

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

Then $\struct {R, +, \circ}$ is a ringoid.

Proof
By definition of ringoid, $\circ_k$ distributes over $+_k$ for all $k = 1, 2, \ldots, n$.

Let $x, y, z \in R$.

Then:

In the same way:


 * $\paren {y + z} \circ x = \paren {y \circ x} + \paren {z \circ x}$

Thus $\circ$ is distributive over $+$.

By definition of ringoid, $\struct {R, +, \circ}$ is a ringoid.