# Definition:Polynomial Ring/Sequences

## Definition

Let $R$ be a commutative ring with unity.

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 $\left({R^{\left({\N}\right)}, \iota, X}\right)$.

That is:

- $R^{\left({\N}\right)}$ is regarded as $R \sqbrk X$
- $\iota : R \to R \sqbrk X$ is the canonical embedding
- $X$ is the Indeterminate

In particular, for $r \in R$ and $n \in \N$:

- $\map \iota r \odot X^n = \sequence {\underbrace{0,\ldots,0}_n,r,0,\ldots}$

is regarded as and written as:

- $r X^n$

## Indeterminate

#### Single indeterminate

Let $\left({S, \iota, X}\right)$ be a polynomial ring over $R$.

The **indeterminate** of $\left({S, \iota, X}\right)$ is the term $X$.

#### Multiple Indeterminates

Let $I$ be a set.

Let $\left({S, \iota, f}\right)$ be a polynomial ring over $R$ in $I$ **indeterminates**.

The **indeterminates** of $\left({S, \iota, f}\right)$ are the elements of the image of the family $f$.

## Notation

It is common to denote a **polynomial ring** $\struct {S, \iota, X}$ over $R$ as $R \sqbrk X$, where $X$ is the indeterminate of $\struct {S, \iota, X}$.

The embedding $\iota$ is then implicit.

## Also see

- Equivalence of Definitions of Polynomial Ring
- Polynomial Ring of Sequences Satisfies Universal Property

## Sources

- 1970: B. Hartley and T.O. Hawkes:
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