# Equivalence of Definitions of Polynomial Ring in Multiple Variables

## Theorem

Let $R$ be a commutative ring with unity.

The following definitions of polynomial ring are equivalent in the following sense:

For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates.

### Definition 1: As the monoid ring on a free monoid on 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 $\left({\left({A, +, \circ}\right), \iota, \left\{ {X_i: i \in I}\right\} }\right)$

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