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

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

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

Let $+$ and $\circ$ denote the standard addition and multiplication of polynomial forms.

Then $\left({A, +, \circ}\right)$ is known as the ring of polynomial forms in $\left\{{X_j: j \in J}\right\}$ over $R$.

Also see

 * Ring of Polynomial Forms is Commutative Ring with Unity