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 $R \left[{\left\{{X_i: i \in I}\right\}}\right]$ be the ring of polynomial forms in $\left\{{X_i: i \in I}\right\}$.

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

 * Equivalence of Definitions of Polynomial Ring