Definition:Polynomial Ring

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One Indeterminate

Let $R$ be a commutative ring with unity.

The set of polynomials over $R$ can be made a ring.


A polynomial ring in one variable is a certain pointed algebra over $R$, that is, an ordered triple $\left({S, \iota, X}\right)$ where:

$S$ is a commutative ring with unity
$\iota : R \to S$ is a unital ring homomorphism, called canonical embedding
$X$ is an element of $S$, called indeterminate

that can be defined in several ways:

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


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.


Definition by Universal Property

A polynomial ring over $R$ is a pointed $R$-algebra $(S, \iota, X)$ that satisfies the following universal property:

For every pointed $R$-algebra $(A, \kappa, a)$ there exists a unique pointed algebra homomorphism $h : S\to A$, called evaluation homomorphism.

This is known as the universal property of a polynomial ring.


Multiple Indeterminates

Let $R$ be a commutative ring with unity.

Let $I$ be a set.


A polynomial ring in $I$ variables is a certain $I$-pointed algebra over $R$, that is, an ordered triple $\left({S, \iota, f}\right)$ where:

$S$ is a commutative ring with unity
$\iota : R \to S$ is a unital ring homomorphism, called canonical embedding
$f : S \to R$ is a family, whose image consists of indeterminates

that can be defined in several ways:

As a 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 $\left({\left({A, +, \circ}\right), \iota, \left\{ {X_i: i \in I}\right\} }\right)$


Terminology

Indeterminates

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 $f$.


Canonical Embedding

Let the ordered triple $(S, \iota, X)$ be a polynomial ring over $R$ in one indeterminate $X$.


The unital ring homomorphism $\iota$ is called the canonical embedding into the polynomial ring.


Multiple Indeterminates

Let $I$ be a set.

Let $(S, \iota, X)$ be a polynomial ring over $R$ in $I$ indeterminates.


The unital ring homomorphism $\iota$ is called the canonical embedding into the polynomial ring.


Notation

It is common to denote a polynomial ring $\left({S, \iota, X}\right)$ over $R$ as $R \left[{X}\right]$, where $X$ is the indeterminate of $\left({S, \iota, X}\right)$.

The embedding $\iota$ is then implicit.


Equivalence of definitions

While, strictly speaking, the above definitions of polynomial ring do define different objects, they can be shown to be isomorphic in a strong sense.

See Equivalence of Definitions of Polynomial Ring.


Also defined as

It is common for an author to define the polynomial ring using a specific construction, and refer to other constructions as a polynomial ring. At $\mathsf{Pr} \infty \mathsf{fWiki}$ we deliberately do not favor any construction. All the more so because at some point it becomes irrelevant.

It is also common to call any ring isomorphic to a polynomial ring a polynomial ring. For the precise meaning of this, see Ring Isomorphic to Polynomial Ring is Polynomial Ring.

Also known as

The polynomial ring in one indeterminate over $R$ is often referred to as the polynomial ring over $R$.

That is, if no reference is given to the number of indeterminates, it is assumed to be $1$.


Also see


Generalizations