Existence of Ordered Dual Basis

Definition
Let $R$ be a commutative ring.

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

Let $\sequence {a_n}$ be an ordered basis of $G$.

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

Let $\sequence {a'_n}$ be the ordered dual basis of $G^*$.

This ordered dual basis $\sequence {a'_n}$ is guaranteed to exist.

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