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 (an) as the the largest integer d such that ad is non-zero.
We define "convolution" of sequences by
Let X denote the sequence
and so on, so that
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 .
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.
- 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.
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].