# Rings In Multiple Variables

Could someone explain to me how to interpret rings written in multiple variables? For example, I am asked to show that $R[[x]][y]$ is a proper subset of $R[y][[x]]$, but I thought that these two were the same. I know that $R[[x]]$ is the ring of formal power series, and I know that $R[x]$ is the ring of polynomials, so I assumed that $R[[x]][y]$ was the ring of formal powers series in x and polynomials in y... but that doesn't seem to be the case. So what would the difference be between $R[[x]][y]$ and $R[y][[x]]$, and why does the order of these matter? Does this generalize with more than two variables, say $R[[x]][y][[z]]$?

Thanks!

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$R[[x]][y]$ is $S[y]$ where $S = R[[x]]$. That is, it's the ring of polynomials in $y$ with coefficients in formal power series in $x$. A typical element of this ring looks like
$$\sum_{i=0}^n \sum_{j=0}^{\infty} (a_{ij} x^j) y^i.$$
On the other hand, $R[y][[x]] = T[[x]]$ where $T = R[y]$. That is, it's the ring of formal power series in $x$ with coefficients in polynomials in $y$. A typical element of this ring looks like
$$\sum_{i=0}^{\infty} \sum_{j=0}^{n_i} (b_{ij} y^j) x^i.$$
As subrings of the ring of formal power series in both $x$ and $y$, can you write down an element of the latter that isn't contained in the former? (Hint: try grouping terms in the first expression to look like the second expression and see what goes wrong.)