Equivalence of Definitions of Polynomial Ring

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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 $R \left[{\N}\right]$ be the monoid ring of $\N$ over $R$.


The polynomial ring over $R$ is the ordered triple $\left({R \left[{\N}\right], \iota, X}\right)$ where:

$X \in R \left[{\N}\right]$ is the standard basis element associated to $1\in \N$.
$\iota : R \to R \left[{\N}\right]$ 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 $R \left[{\left\{{X_i: i \in I}\right\}}\right]$ 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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