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 $R[[x]][y][[z]]$?