Polynoomalgebra

Polynoomalgebra, or polynomial algebra, is the algebra of polynomials with coefficients in a ring R in a set of indeterminates X1, X2, ..., Xn. It is denoted R[X1,...,Xn] and consists of all finite sums of terms of the form a X1^k1 X2^k2 ... Xn^kn with a in R and nonnegative integers ki. As an R-algebra, it contains R as a subring and the indeterminates X1,...,Xn generate it freely.

A defining feature is its universal property: for any commutative R-algebra A and any map from the set {X1,...,Xn} into A, there exists a unique R-algebra homomorphism extending that map. This makes R[X1,...,Xn] the free commutative R-algebra on n generators.

Structure and basic properties follow from the nature of the coefficients and exponents. If R is a field F, F[X1,...,Xn] is a graded ring by total degree, and its homogeneous components give a natural grading. For n = 1, F[x] is a principal ideal domain and a Euclidean domain; for n > 1 it remains a UFD but loses these larger-degree properties.

General features include that polynomial algebras are Noetherian whenever R is Noetherian (Hilbert basis theorem), and their Krull dimension satisfies dim R[X1,...,Xn] = dim R + n. If R is an integral domain, so is the polynomial ring. Polynomial algebras underpin many areas of algebra and geometry, serving as the coordinate rings of affine spaces and as a framework for ideals, varieties, and computational methods such as Groebner bases.