Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set

Theorem
Let  $$\left({R, +,  \circ}\right)$$ be a   commutative ring with unity.

Let $$R \left[\{{X_j:j\in J}\}\right]$$ be the set of all polynomials over $$R$$ in the indeterminates $$\{{X_j:j\in J}\}$$.

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

For two polynomials $$\displaystyle f = \sum_{k\in Z} a_k \mathbf X^k,\ g = \sum_{k\in Z} b_k \mathbf X^k$$ in $$R \left[\{{X_j:j\in J}\}\right]$$, define the sum:


 * $\displaystyle f \oplus g = \sum_{k\in Z} \left({a_k + b_k}\right)\mathbf X^k$

and the product


 * $\displaystyle f \otimes g = \sum_{k\in Z} c_i \mathbf X^k$

where $$\displaystyle c_k = \sum_{j+l = k} f \left({\mathbf X^j}\right) \circ g \left({\mathbf X^l}\right)$$.

Then $$\left({R \left[\{{X_j:j\in J}\}\right], \oplus, \otimes}\right)$$ is a commutative ring with unity.