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.


[edit] 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].

[edit] 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].

[edit] Properties

[edit] 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].

[edit] References

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