Editing Polynomial ring (section)

=== Infinitely many variables ===
One slight generalization of polynomial rings is to allow for infinitely many indeterminates.  Each monomial still involves only a finite number of indeterminates (so that its degree remains finite), and each polynomial is a still a (finite) linear combination of monomials.  Thus, any individual polynomial involves only finitely many indeterminates, and any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. This generalization has the same property of usual polynomial rings, of being the [[free commutative algebra]], the only difference is that it is a [[free object]] over an infinite set.

One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than the [[power series ring#Power series in several variables|ring of power series in infinitely many variables]]. Such a ring is used for constructing the [[ring of symmetric functions]] over an infinite set.