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.

This article, or a section of it, needs explaining. In particular: this statement has to be made more precise 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.

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.

This article, or a section of it, needs explaining. In particular: this statement has to be made more precise 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.

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 case You 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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