Definition:Polynomial Ring
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 $\struct {S, \iota, X}$ 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 $\left \langle {0, 1, 0, \ldots}\right \rangle$.
 | 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 $\left({R^{\left({\N}\right)}, \iota, X}\right)$.
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.
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 $\struct {S, \iota, f}$ 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 $R \sqbrk {\family {X_i: i \in I} }$ 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. |
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 $\struct {S, \iota, X}$ over $R$ as $R \sqbrk X$, where $X$ is the indeterminate of $\struct {S, \iota, X}$.
The embedding $\iota$ is then implicit.
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
- Results about polynomial rings can be found here.
Generalizations
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polynomial ring