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


 * $$\left(a_0, a_1, \ldots, a_n, \ldots \right) \,$$

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


 * $$(a+b)_n = a_n + b_n .\,$$

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


 * $$(a \star b)_n = \sum_{i=0}^n a_i b_{n-i} .\,$$

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

Let X denote the sequence


 * $$X = (0,1,0,\ldots) .\,$$

We have


 * $$X^2 = X \star X = (0,0,1,0,\ldots) \,$$
 * $$X^3 = X \star X \star X = (0,0,0,1,0,\ldots) \,$$

and so on, so that


 * $$ \left(a_0, a_1, \ldots, a_n, \ldots \right) = \sum_n a_n X^n ,\,$$

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 $$R[X]$$.

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

 * 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 $$\deg(fg) = \deg(f) + \deg(g)$$.
 * 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 $$f:A \rarr B$$ is a ring homomorphism then there is a homomorphism, also denoted by f, from $$A[X] \rarr B[X]$$ 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


 * $$R[X_1,X_2,\ldots,X_n] = R[X_1][X_2]\ldots[X_n] ,\,:$$

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

We see that there are natural isomorphisms


 * $$R[X_1][X_2] \equiv R[X_1,X_2] \equiv R[X_2][X_1] .\,$$

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