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#''S'' is invariant under any permutation of the indeterminates, and
#the 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'' > 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 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'' ≥ 0, the element ''e''<sub>''k''</sub> ∈ Λ<sub>''R''</sub> is defined as the formal sum of all products of ''k'' distinct indeterminates, which is clearly homogeneous of degree ''k''.
==== As an algebraic limit ====
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