polynomial ring

1 Polynomial rings in one variable

Let R be a ring. The polynomial ringMathworldPlanetmath over R in one variable X is the set R[X] of all sequencesPlanetmathPlanetmath in R with only finitely many nonzero terms. If (a0,a1,a2,a3,) is an element in R[X], with an=0 for all n>N, then we usually write this element as


Elements of R[X] are called polynomialsMathworldPlanetmath in the indeterminate X with coefficients in R. The ring elements a0,,aN are called coefficients of the polynomial, and the degree of a polynomial is the largest natural numberMathworldPlanetmath N for which aN0, 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 -.

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

n=0NanXn+n=0NbnXn = n=0N(an+bn)Xn (1)
n=0NanXnn=0NbnXn = n=02N(k=0nakbn-k)Xn (2)

R[X] is a graded ringMathworldPlanetmath under these operationsMathworldPlanetmath, with the monomials of degree exactly n comprising the nth graded componentMathworldPlanetmath of R[X]. The zero elementMathworldPlanetmath of R[X] is the polynomial whose coefficients are all 0, and when R has a multiplicative identityPlanetmathPlanetmath 1, the polynomial whose coefficients are all 0 except for a0=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]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]R[X][Z][Y], and in any finite number of variables, we have inductively R[X1,X2,,Xn]:=R[X1,,Xn-1][Xn]=R[X1][X2][Xn], with monomials and degrees defined in analogyMathworldPlanetmath to the two variable case. In any number of variables, a polynomial ring is a graded ring with nth 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={a1,,an} of E(M), set R[A]:=R[a1,,an]. Any two elements A,BE(M) satisfying AB give rise to the relationship R[A]R[B] if we consider R[A] to be embedded in R[B] in the obvious way. The union of the rings {R[A]:AE(M)} (or, more formally, the categorical direct limitMathworldPlanetmath of the direct systemMathworldPlanetmath of rings {R[A]:AE(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