Bounded Degree in Ring of Symmetric Functions The Ring of Symmetric Functions over a commutative ring $R$, $\Lambda$, is the subring of the ring of formal power series $R[[x_1, x_2, \dots]]$ such that $f \in \Lambda$ if $f$ is invariant under every permutation of of the indeterminants and the degrees of the monomials occurring in $f$ are bounded.
Why do we need this second condition?  I know the second condition gives that $\Lambda \cong R[\sigma_1,\sigma_2, \dots]$, the polynomial ring in the infinite indeterminants $\sigma_i$, but is there another reason we require this?  Thanks!
 A: *

*One reason for considering $\Lambda$ is that many systems of symmetric polynomials form an honest $R$-basis of $\Lambda$, for example elementary symmetric, monomial symmetric, power symmetric, and Schur functions. The base change matrices are very important in combinatorics.

*On the other hand, it is not true that $R[[x_i]]^{S_\infty}$ is irrelevant. For example a formula like
$$
\log(1+e_1+e_2+\ldots)=\log\prod_i(1+x_i)=\sum_i\log(1+x_i)=\sum_{n\ge1}\frac{(-1)^{n-1}}np_n
$$
is an identity in the latter ring. By comparing homogeneous components, it can be transformed into a sequence of identities in $\Lambda$, though.

*So the answer to your question might be that $\Lambda$ has strictly more structure than $R[[x_i]]^{S_\infty}$, namely a grading, and that most objects occuring in nature respect this structure, i.e., have a degree.

A: Because we don't want to include power series like
$$\frac{1}{1-\sigma_1}=1+\sigma_1+\sigma_1^2+\sigma_1^3+\cdots.$$ We barely even want power series at all; we're just using them as a notational convenience to allow ourselves to talk about $\sigma_1=x_1+x_2+\cdots$ as a stand-in for "$\sigma_1=x_1+\cdots+x_N$ where $N$ is an undetermined quantity that we arbitrarily increase whenever necessary".
