Unique Representation in Polynomial Forms

Theorem
Let $$(R,M,f)$$ be a polynomial in the indeterminates $$\{X_j:j\in J\}$$.

For $$r\in R$$, $$\mathbf X^k\in M$$, let $$r\mathbf X^k$$ denote the polynomial that takes the value $$r$$ on $$\mathbf X^k$$ and zero on all other mononomials.

Let $$Z$$ denote the set of all multiindices indexed by $$J$$.

Then every polynomial can be uniquely written as a sum


 * $\sum_{k\in Z}r_k\mathbf X^k$

with only finitely many non-zero terms.