Ring of symmetric functions: Difference between revisions

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==== 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]] ''R''[[''X''₁,''X''₂,…]] over ''R'' in infinitely (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_''R'' as its subring consisting of those power series ''S'' that satisfy

#''S'' is invariant under any permutation of the indeterminates, and

#the degrees of the monomials occurring in ''S'' are bounded.