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Property 2 is the essence of the [[fundamental theorem of symmetric polynomials]]. It immediately implies some other properties:
* The subring of Λ<sub>''R''</sub> generated by its elements of degree at most ''n'' is isomorphic to the ring of symmetric polynomials over ''R'' in ''n'' variables;
* The [[Hilbert–Poincaré series]] of Λ<sub>''R''</sub> is <math>\textstyle\prod_{i=1}^\infty\frac1{1-t^i}</math>, the [[
* For every ''n'' > 0, the ''R''-module formed by the homogeneous part of Λ<sub>''R''</sub> of degree ''n'', modulo its intersection with the subring generated by its elements of degree strictly less than ''n'', is [[free module|free]] of rank 1, and (the image of) ''e''<sub>''n''</sub> is a generator of this ''R''-module;
* For every family of symmetric functions (''f''<sub>''i''</sub>)<sub>''i''>0</sub> in which ''f''<sub>''i''</sub> is homogeneous of degree ''i'' and gives a generator of the free ''R''-module of the previous point (for all ''i''), there is an alternative isomorphism of graded ''R''-algebras from ''R''[''Y''<sub>1</sub>,''Y''<sub>2</sub>, ...] as above to Λ<sub>''R''</sub> that sends ''Y''<sub>''i''</sub> to ''f''<sub>''i''</sub>; in other words, the family (''f''<sub>''i''</sub>)<sub>''i''>0</sub> forms a set of free polynomial generators of Λ<sub>''R''</sub>.
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