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

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

Let $$R \left[{X}\right] = \left\{{a_0 + \cdots + a_n X^n: a_i \in R, n \in \N}\right\}$$ be the set of all polynomials over $$R$$.

For two polynomials $$\displaystyle f = \sum_{i=0}^\infty a_i X^i,\ g = \sum_{i=1}^\infty b_i X^i$$ in $$R \left[{X}\right]$$, define the sum:


 * $\displaystyle f \oplus g = \sum_{i=0}^\infty \left({a_i + b_i}\right) X^i$

and the product


 * $\displaystyle f \otimes g = \sum_{i=0}^\infty c_i X^i$

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

Then $$\left({R \left[{X}\right], \oplus, \otimes}\right)$$ is a ring.