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 (a_n) as the the largest integer d such that a_d 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 a_n 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 $f:A\rightarrow B$ is a ring homomorphism then there is a homomorphism, also denoted by f, from $A[X]\rightarrow B[X]$ 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].

References

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