Basis for Finite Submodule of Function Space

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

Let $$A$$ be a set.

For each $$a \in A$$, let $$f_a: A \to R$$ be defined as $$\forall x \in A: f_a \left({x}\right) = \begin{cases} 1 & : x = a \\ 0 & : x \ne a \end{cases} $$.

Then $$B = \left\{{f_a: a \in A}\right\}$$ is a basis of the Finite Submodule of Function Space $$R^{\left({A}\right)}$$.