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

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

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 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.

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 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