polynomial ring

1 Polynomial rings in one variable

Let $R$ be a ring. The polynomial ring over $R$ in one variable $X$ is the set $R[X]$ of all sequences in $R$ with only finitely many nonzero terms. If $(a_0,a_1,a_2,a_3,\mathrm{\dots })$ is an element in $R[X]$, with $a_n=0$ for all $n>N$, then we usually write this element as

$$\sum _{n=0}^N{a}_n{X}^n=a_0+a_{1}X+a_2{X}^2+a_3{X}^3+\mathrm{\cdots }+a_N{X}^N.$$

Elements of $R[X]$ are called polynomials in the indeterminate $X$ with coefficients in $R$. The ring elements $a_0,\mathrm{\dots },a_N$ are called coefficients of the polynomial, and the degree of a polynomial is the largest natural number $N$ for which $a_N\ne 0$, if such an $N$ exists. When a polynomial has all of its coefficients equal to $0$, its degree is usually considered to be undefined, although some people adopt the convention that its degree is $-\mathrm{\infty }$.

A monomial is a polynomial with exactly one nonzero coefficient. Similarly, a binomial is a polynomial with exactly two nonzero coefficients, and a trinomial is a polynomial with exactly three nonzero coefficients.

Addition and multiplication of polynomials is defined by

${\displaystyle \sum _{n=0}^N}{a}_n{X}^n+{\displaystyle \sum _{n=0}^N}{b}_n{X}^n$ $=$ ${\displaystyle \sum _{n=0}^N}(a_n+b_n)X^n$ (1)

${\displaystyle \sum _{n=0}^N}{a}_n{X}^n\cdot {\displaystyle \sum _{n=0}^N}{b}_n{X}^n$ $=$ ${\displaystyle \sum _{n=0}^{2N}}\left({\displaystyle \sum _{k=0}^n}{a}_k{b}_{n-k}\right)X^n$ (2)

$R[X]$ is a $\mathbb{Z}$–graded ring under these operations, with the monomials of degree exactly $n$ comprising the $n^{\mathrm{th}}$ graded component of $R[X]$. The zero element of $R[X]$ is the polynomial whose coefficients are all $0$, and when $R$ has a multiplicative identity $1$, the polynomial whose coefficients are all $0$ except for $a_0=1$ is a multiplicative identity for the polynomial ring $R[X]$.

2 Polynomial rings in finitely many variables

The polynomial ring over $R$ in two variables $X,Y$ is defined to be $R[X,Y]:=R[X][Y]\cong R[Y][X]$. Elements of $R[X,Y]$ are called polynomials in the indeterminates $X$ and $Y$ with coefficients in $R$. A monomial in $R[X,Y]$ is a polynomial which is simultaneously a monomial in both $X$ and $Y$, when considered as a polynomial in $X$ with coefficients in $R[Y]$ (or as a polynomial in $Y$ with coefficients in $R[X]$). The degree of a monomial in $R[X,Y]$ is the sum of its individual degrees in the respective indeterminates $X$ and $Y$ (in $R[Y][X]$ and $R[X][Y]$), and the degree of a polynomial in $R[X,Y]$ is the supremum of the degrees of its monomial summands, if it has any.

In three variables, we have $R[X,Y,Z]:=R[X,Y][Z]=R[X][Y][Z]\cong R[X][Z][Y]\cong \mathrm{\cdots }$, and in any finite number of variables, we have inductively $R[X_1,X_2,\mathrm{\dots },X_n]:=R[X_1,\mathrm{\dots },X_{n-1}][X_n]=R[X_1][X_2]\mathrm{\cdots }[X_n]$, with monomials and degrees defined in analogy to the two variable case. In any number of variables, a polynomial ring is a graded ring with $n^{\mathrm{th}}$ graded component equal to the $R$-module generated by the monomials of degree $n$.

3 Polynomial rings in arbitrarily many variables

For any nonempty set $M$, let $E(M)$ denote the set of all finite subsets of $M$. For each element $A=\{a_1,\mathrm{\dots },a_n\}$ of $E(M)$, set $R[A]:=R[a_1,\mathrm{\dots },a_n]$. Any two elements $A,B\in E(M)$ satisfying $A\subset B$ give rise to the relationship $R[A]\subset R[B]$ if we consider $R[A]$ to be embedded in $R[B]$ in the obvious way. The union of the rings $\{R[A]:A\in E(M)\}$ (or, more formally, the categorical direct limit of the direct system of rings $\{R[A]:A\in E(M)\}$) is defined to be the ring $R[M]$.

Title	polynomial ring
Canonical name	PolynomialRing
Date of creation	2013-03-22 11:52:27
Last modified on	2013-03-22 11:52:27
Owner	djao (24)
Last modified by	djao (24)
Numerical id	10
Author	djao (24)
Entry type	Definition
Classification	msc 11C08
Classification	msc 12E05
Classification	msc 13P05
Classification	msc 17B66
Classification	msc 16W10
Classification	msc 70G65
Classification	msc 17B45
Related topic	AlgebraicGeometry
Related topic	RationalFunction
Defines	polynomial
Defines	monomial
Defines	binomial
Defines	trinomial
Defines	degree