Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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


share|cite|improve this question
up vote 2 down vote accepted

$R[[x]][y]$ 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.)

share|cite|improve this answer
Thanks! This was a crystal clear explanation :) – Nizbel99 Nov 3 '12 at 18:31

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.