Existence of Ordered Dual Basis

Definition
Let $R$ be a commutative ring.

Let $\left({G, +_G, \circ}\right)_R$ be an $n$-dimensional module over $R$.

Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$.

Let $G^*$ be the algebraic dual of $G$.

Let $\left \langle {a'_n} \right \rangle$ be the ordered dual basis of $G^*$.

This ordered dual basis $\left \langle {a'_n} \right \rangle$ is guaranteed to exist.

Proof
Since $\left\{{1_R}\right\}$ is a basis of the $R$-module $R$, by Product of Linear Transformations this basis as described exists.