Difference between $R[x]$ and $R[[x]]$

Question

I am watching Richard E. BORCHERDS lectures on Rings and Modules 21 Formal power series and have a question about relationship between $R[x]$ and $R[[x]]$.

Richard just say $R[[x]]$ as formal power series and state $a_0 + a_1 x + a_2 x^2+ \cdots, a_i \in R$, and denote $R[[x]] = \lim_{\leftarrow} R[x]/(x^m)$ inverse limit of $R[x]/(x^m)$. [Can you explain this in more detail? I watch them over and over again but can not understand its meaning.]

But I am confused (maybe because I am not familiar with inverse limits). I know $R[x]$ is a polynomial ring $R[x]= \{ a_0 + a_1 x + \cdots + a_n x^n | a_i \in R\}$, What is the relation between them? My naive guess is $R[[x]]$ is infinite sums, but $R[x]$ is finite sums. What makes them different in many aspects?

I know $R$ is UFD then $R[x]$ is UFD but $R[[x]]$ is not UFD. And $R$ is Noetherian then $R[x]$ is Noetherian but $R[[x]]$ is also Noetherian i.e., all the ideals are finitely generated.

So what makes $R[x]$ and $R[[x]]$ similar and different approaching some mathematical properties?

Answer 1

Inverse limit of the rings $E_m=R[x]/(x^m)$ just means that if $m>n$ then there is a natural map $E_m\to E_n$ and that we consider the set of sequences $(a_m)_{m\ge 1}$ with $ a_m\in E_m$ and such that $m>n\implies a_m = a_n\bmod (x^n)$. This set is naturally a ring: $(a_m)+(b_m)= (a_m+b_m)$, $(a_m)(b_m)=(a_mb_m)$ (addition and multiplication in each ring $E_m$) and it is naturally isomorphic to the ring of formal power series. For a power series $\sum_{k\ge 0} c_k x^k$, letting $a_m=\sum_{k=0}^{m-1} c_k x^k$ then $(a_m)$ is an element of $\varprojlim R[x]/(x^m)$ and they are all of this form.

Answer 2

$R[X]$ is the polynomial ring as you know it. $R[[X]]$ is the ring of formal power series, we define $$ R[[X]] = \varprojlim_n R[X]/(X)^n.$$ Elements in this ring are sequences $(f_n+(X)^n)_n$ of residue classes of polynomials $f_n \in R[X]$ with the property that if $m\ge n$ then $f_m + (X)^n = f_n + (X)^n$ and so it is regarded as a subring of $\prod_n R[X]/(X)^n$. What is the intuition here? Well, a power series can be thought of as a sequence of polynomials of finite degree increasing degree where we chop off the series at a given point. But we cant take any arbitrary sequence, the condition we want for this sequence to represent a power series is that the terms must "agree" i.e. if I look take the series up to its $m$-th term and chop it down to its $n$-th term where $m\ge n$ then I should get the sequences $n$-th term. But this is precisely the condition specified in the definition.