# Equivalence of Definitions of Polynomial Ring

## One Variable

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 a Ring of Sequences

Let $R^{\left({\N}\right)}$ be the ring of sequences of finite support over $R$.

Let $\iota : R \to R^{\left({\N}\right)}$ be the mapping defined as:

$\iota \left({r}\right) = \left \langle {r, 0, 0, \ldots}\right \rangle$.

Let $X$ be the sequence $\left \langle {0, 1, 0, \ldots}\right \rangle$.

The polynomial ring over $R$ is the ordered triple $\left({R^{\left({\N}\right)}, \iota, X}\right)$.

### Definition 2: As a Monoid Ring on the Natural Numbers

Let $\N$ denote the additive monoid of natural numbers.

Let $R \sqbrk \N$ be the monoid ring of $\N$ over $R$.

The polynomial ring over $R$ is the ordered triple $\struct {R \sqbrk \N, \iota, X}$ where:

$X \in R \sqbrk \N$ is the standard basis element associated to $1 \in \N$
$\iota : R \to R \sqbrk \N$ is the canonical mapping.

## Multiple Variables

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 $R \sqbrk {\family {X_i: i \in I} }$ be the ring of polynomial forms in $\family {X_i: i \in I}$.

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