Editing Ring of symmetric functions (section)

==== As a ring of formal power series ====

The easiest (though somewhat heavy) construction starts with the ring of [[Formal power series#Power series in several variables|formal power series]] <math>R[[X_1,X_2,...]]</math> over ''R'' in infinitely ([[countably infinite|countably]]) many indeterminates; the elements of this [[power series]] ring are formal infinite sums of terms, each of which consists of a coefficient from ''R'' multiplied by a [[monomial]], where each monomial is a product of finitely many finite powers of indeterminates. One defines Λ<sub>''R''</sub> as its subring consisting of those power series ''S'' that satisfy 
#''S'' is invariant under any permutation of the indeterminates, and
#the [[degree of a polynomial|degrees]] of the monomials occurring in ''S'' are bounded.
Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixed degree, rather than to sum terms of all possible degrees. Allowing this is necessary because an element that contains for instance a term ''X''<sub>1</sub> should also contain a term ''X''<sub>''i''</sub> for every ''i''&nbsp;>&nbsp;1 in order to be symmetric. Unlike the whole power series ring, the subring Λ<sub>''R''</sub> is graded by the total degree of monomials: due to condition&nbsp;2, every element of Λ<sub>''R''</sub> is a finite sum of [[Homogeneous polynomial|homogeneous]] elements of Λ<sub>''R''</sub> (which are themselves infinite sums of terms of equal degree). For every ''k''&nbsp;≥&nbsp;0, the element ''e''<sub>''k''</sub>&nbsp;∈&nbsp;Λ<sub>''R''</sub> is defined as the formal sum of all products of ''k'' distinct indeterminates, which is clearly homogeneous of degree ''k''.