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

Definition
Let $R$ be a commutative ring with unity.

Let $I$ be a set.

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

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

The polynomial ring in $I$ indeterminates over $R$ is the ordered triple $\left(\left({A, +, \circ}\right), \iota, \left\{{X_i: i \in I}\right\}\right)$

Also see

 * Ring of Polynomial Forms is Commutative Ring with Unity