# 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.

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### 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 $\left \langle {0, 1, 0, \ldots}\right \rangle$.

This article, or a section of it, needs explaining.In particular: Please clarify the role of $X$. This does not look like a ring to me. What is the multiplication? --Wandynsky (talk) 17:17, 30 July 2021 (UTC)
What is not clear? $R^{\left({\N}\right)}$ is a ring. --Usagiop (talk) 19:16, 28 September 2022 (UTC) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

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.

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### 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 $\struct {\struct {A, +, \circ}, \iota, \family {X_i: i \in I} }$

This definition needs to be completed.In particular: define the inclusion and indeterminates in this caseYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{DefinitionWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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