polynomial ring

A polynomial ring is the ring you get by adjoining one or more 'free' indeterminate symbols to a base ring and letting them be added and multiplied with the existing elements, subject to no relations at all. It is the most basic way to grow a ring, and the universal home for substitution: anything you can plug in for the variables, you can.

Given a commutative ring R, the polynomial ring R[x] consists of all finite formal sums a_n x^n + ... + a_1 x + a_0 with coefficients a_i in R, added coefficientwise and multiplied by the usual convolution rule. One iterates to get R[x_1, ..., x_n] in several variables. The defining universal property is that R[x] is the free commutative R-algebra on one generator: a homomorphism out of R[x] is determined freely by a choice of image for x.

Polynomial rings are the workhorses of algebra. Hilbert's basis theorem says R[x] is Noetherian whenever R is; Gauss's lemma says R[x] is a UFD whenever R is. Over a field, k[x] is a Euclidean domain and hence a PID, but k[x, y] is neither — the ideal (x, y) needs two generators. Geometrically, k[x_1, ..., x_n] is the coordinate ring of affine n-space, so its ideals encode systems of polynomial equations and their solution sets.