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 $R$.