Definition:Module on Cartesian Product

Theorem
Let $$\left({R, +_R, \times_R}\right)$$ be a ring.

Let $$n \in \N^*$$.

Let $$+: R^n \times R^n \to R^n$$ be defined as $$\left({\alpha_1, \ldots, \alpha_n}\right) + \left({\beta_1, \ldots, \beta_n}\right) = \left({\alpha_1 +_R \beta_1, \ldots, \alpha_n +_R \beta_n}\right)$$

Let $$\times: R \times R^n \to R^n$$ be defined as $$\lambda \times \left({\alpha_1, \ldots, \alpha_n}\right) = \left({\lambda \times_R \alpha_1, \ldots, \lambda \times_R \alpha_n}\right)$$

Then $$\left({R^n, +, \times}\right)_R$$ is an $R$-module.

This will be referred to as the $$R$$-module $$R^n$$.

If $$R$$ is a ring with unity, $$\left({R^n, +, \times}\right)_R$$ is a unitary $R$-module.

Proof
This is a special case of Module of All Mappings, where $$S$$ is the set $$\left[{1 \,. \, . \, n}\right] \subset \N^*$$.

It is also a special case of a Module Product where each of the $$G_k$$ is the $R$-module $$R$$.