# Polynomial ring

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

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

We define the *degree* of a non-zero sequence (*a*_{n}) as the the largest integer *d* such that
*a*_{d} is non-zero.

We define "convolution" of sequences by

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

Let *X* denote the sequence

We have

and so on, so that

which makes sense as a finite sum since only finitely many of the *a*_{n} are non-zero.

The ring defined in this way is denoted .

### 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

- The polynomial ring
*R*[*X*] is an algebra over*R*. - If
*R*is commutative then so is*R*[*X*]. - If
*R*is an integral domain then so is*R*[*X*].- In this case the degree function satisfies .

- If
*R*is a unique factorisation domain then so is*R*[*X*]. *Hilbert's basis theorem*: If*R*is a Noetherian ring then so is*R*[*X*].- If
*R*is a field, then*R*[*X*] is a Euclidean domain.

- If is a ring homomorphism then there is a homomorphism, also denoted by
*f*, from 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

but a more general construction which allows the construction of polynomials in any set of variables is to follow the initial construction by taking *S* to be the Cartesian power and then to consider the *R*-valued functions on *S* with finite support.

We see that there are natural isomorphisms

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

## References

- Serge Lang (1993).
*Algebra*, 3rd ed. Addison-Wesley, 97-98. ISBN 0-201-55540-9.