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Another construction of Λ<sub>''R''</sub> takes somewhat longer to describe, but better indicates the relationship with the rings ''R''[''X''<sub>1</sub>,…,''X''<sub>''n''</sub>]<sup>'''S'''<sub>''n''</sub></sup> of symmetric polynomials in ''n'' indeterminates. For every ''n'' there is a surjective [[ring homomorphism]] ρ<sub>''n''</sub> from the analoguous ring ''R''[''X''<sub>1</sub>,…,''X''<sub>''n''+1</sub>]<sup>'''S'''<sub>''n''+1</sub></sup> with one more indeterminate onto ''R''[''X''<sub>1</sub>,…,''X''<sub>''n''</sub>]<sup>'''S'''<sub>''n''</sub></sup>, defined by setting the last indeterminate ''X''<sub>''n''+1</sub> to 0. Although ρ<sub>''n''</sub> has a non-trivial kernel, the nonzero elements of that kernel have degree at least <math>n+1</math> (they are multiples of ''X''<sub>1</sub>''X''<sub>2</sub>…''X''<sub>''n''+1</sub>). This means that the restriction of ρ<sub>''n''</sub> to elements of degree at most ''n'' is a bijective linear map, and ρ<sub>''n''</sub>(''e''<sub>''k''</sub>(''X''<sub>1</sub>,…,''X''<sub>''n''+1</sub>)) = ''e''<sub>''k''</sub>(''X''<sub>1</sub>,…,''X''<sub>''n''</sub>) for all ''k'' ≤ ''n''. The inverse of this restriction can be extended uniquely to a ring homomorphism φ<sub>''n''</sub> from ''R''[''X''<sub>1</sub>,…,''X''<sub>''n''</sub>]<sup>'''S'''<sub>''n''</sub></sup> to ''R''[''X''<sub>1</sub>,…,''X''<sub>''n''+1</sub>]<sup>'''S'''<sub>''n''+1</sub></sup>, as follows for instance from the [[fundamental theorem of symmetric polynomials]]. Since the images φ<sub>''n''</sub>(''e''<sub>''k''</sub>(''X''<sub>1</sub>,…,''X''<sub>''n''</sub>)) = ''e''<sub>''k''</sub>(''X''<sub>1</sub>,…,''X''<sub>''n''+1</sub>) for ''k'' = 1,…,''n'' are still [[algebraically independent]] over ''R'', the homomorphism φ<sub>''n''</sub> is injective and can be viewed as a (somewhat unusual) inclusion of rings. The ring Λ<sub>''R''</sub> is then the "union" ([[direct limit]]) of all these rings subject to these inclusions. Since all φ<sub>''n''</sub> are compatible with the grading by total degree of the rings involved, Λ<sub>''R''</sub> obtains the structure of a graded ring.
This construction differs slightly from the one in (Macdonald, 1979). That construction only uses the surjective morphisms ρ<sub>''n''</sub> without mentioning the injective morphisms φ<sub>''n''</sub>: it constructs the homogeneous components of Λ<sub>''R''</sub> separately, and equips their direct sum with a ring structure using the ρ<sub>''n''</sub>. It is also observed that the result can be described as an [[inverse limit]] in the category of ''graded'' rings. That description however somewhat obscures an important property typical for a ''direct'' limit of injective morphisms, namely that every individual element (symmetric function) is already faithfully represented in some object used in the limit construction, here a ring ''R''[''X''<sub>1</sub>,…,''X''<sub>''d''</sub>]<sup>'''S'''<sub>''d''</sub></sup>. It suffices to take for ''d'' the degree of the symmetric function, since the part in degree ''d''
=== Defining individual symmetric functions ===
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