# Polynomial ring

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In algebra, the polynomial ring over a ring is a construction of a ring which formalises the polynomials of elementary algebra.

## Construction of the polynomial ring

Let R be a ring. Consider the R-module of sequences

${\displaystyle \left(a_{0},a_{1},\ldots ,a_{n},\ldots \right)\,}$

which have only finitely many non-zero terms, under pointwise addition

${\displaystyle (a+b)_{n}=a_{n}+b_{n}.\,}$

We define the degree of a non-zero sequence (an) as the the largest integer d such that ad is non-zero.

We define "convolution" of sequences by

${\displaystyle (a\star b)_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}.\,}$

Convolution is a commutative, associative operation on sequences which is distributive over addition.

Let X denote the sequence

${\displaystyle X=(0,1,0,\ldots ).\,}$

We have

${\displaystyle X^{2}=X\star X=(0,0,1,0,\ldots )\,}$
${\displaystyle X^{3}=X\star X\star X=(0,0,0,1,0,\ldots )\,}$

and so on, so that

${\displaystyle \left(a_{0},a_{1},\ldots ,a_{n},\ldots \right)=\sum _{n}a_{n}X^{n},\,}$

which makes sense as a finite sum since only finitely many of the an are non-zero.

The ring defined in this way is denoted ${\displaystyle R[X]}$.

### Alternative points of view

We can view the construction by sequences from various points of view

We may consider the set of sequences described above as the set of R-valued functions on the set N of natural numbers (including zero) and defining the support of a function to be the set of arguments where it is non-zero. We then restrict to functions of finite support under pointwise addition and convolution.

We may further consider N to be the free monoid on one generator. The functions of finite support on a monoid M form the monoid ring R[M].

## Properties

• If ${\displaystyle f:A\rightarrow B}$ is a ring homomorphism then there is a homomorphism, also denoted by f, from ${\displaystyle A[X]\rightarrow B[X]}$ which extends f. Any homomorphism on A[X] is determined by its restriction to A and its value at X.

## Multiple variables

The polynomial ring construction may be iterated to define

${\displaystyle R[X_{1},X_{2},\ldots ,X_{n}]=R[X_{1}][X_{2}]\ldots [X_{n}],\,:}$

but a more general construction which allows the construction of polynomials in any set of variables ${\displaystyle \{X_{\lambda }:\lambda \in \Lambda \}}$ is to follow the initial construction by taking S to be the Cartesian power ${\displaystyle \mathbf {N} ^{\Lambda }}$ and then to consider the R-valued functions on S with finite support.

We see that there are natural isomorphisms

${\displaystyle R[X_{1}][X_{2}]\equiv R[X_{1},X_{2}]\equiv R[X_{2}][X_{1}].\,}$

We may also view this construction as taking the free monoid S on the set Λ and then forming the monoid ring R[S].