Standard Ordered Basis is Basis

Theorem
Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $n$ be a positive integer.

For each $j \in \left[{1 \,. \, . \, n}\right]$, let $e_j$ be the ordered $n$-tuple of elements of $R$ whose $j$th entry is $1_R$ and all of whose other entries is $0_R$.

Then $\left \langle {e_n} \right \rangle$ is an ordered basis of the $R$-module $R^n$.

This ordered basis is called the standard ordered basis of $R^n$.

The corresponding set $\left\{{e_1, e_2, \ldots, e_n}\right\}$ is called the standard basis of $R^n$.