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

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.