Let $k=\mathbb{C}$ or $\mathbb{R}$, and let k{x} denote the ring of power series with appropriate coefficients that are convergent around 0. Check that k{x} is a local ring.
I have a similar question:
https://math.stackexchange.com/questions/765780/ring-of-formal-power-series-is-local-ring
Could someone give an example of what k{x} is for $\mathbb{C}$ and $\mathbb{R}$? I do not really understand what k{x} would even be.
For the proof, I know that a ring is local iff it has a unique maximal ideal. Would I want to use this to prove the statement?
Best Answer
The ring (actually $\mathbb C$-algebra) $\mathbb C\{x\}$ is the subring of $\mathbb C[[x]]$ consisting of those power series $\sum a_n x^n$ which represent the development at zero of an analytic function defined on some neighbourhood of zero. Notice that the neighbourhood in question may vary from power series to power series.
While the ring of formal power series $k[[x]]$ may be defined for an arbitrary field (or even ring) $k$, the ring $\mathbb C\{x\}$ depends for its definition on the topology of $\mathbb C$.
To simplify things outrageously, $\mathbb C\{x\}$ is in the realm of analysis and $\mathbb C[[x]]$ in that of algebra.
The great analyst Hadamard has given a wonderful criterion for a power series $\sum a_n x^n \in \mathbb C[[x]]$ to belong to $\mathbb C\{x\}$: $$\sum a_n x^n \in \mathbb C\{x\} \subset \mathbb C[[x]] \iff \limsup (|a_n|^{1/n} )\lt \infty $$
And now, back to your actual question!
The ring $\mathbb C\{x\}$ has a maximal ideal $\mathfrak m$ consisting in the power series $\sum a_n x^n$ with $a_0=0$.
That ring is local because every converging power series $\sum a_n x^n \notin \mathfrak m$, that is with $a_0\neq 0$, is invertible, so that $\mathfrak m$ is the only maximal ideal in the ring .
Indeed $\sum a_n x^n \notin \mathfrak m$ is invertible in $\mathbb C[[x]]$ [by the (virtual) answer to your preceding question :-)] and its inverse in $\mathbb C[[x]]$ can be shown to actually be in $\mathbb C\{x\}$ too.
(Everything I wrote above is still true if you replace $\mathbb C$ by $\mathbb R$)
(Non-) example
Hadamard's theorem trivially shows that the formal power series $\sum n^nx^n \in \mathbb C[[x]]$ is not a convergent power series: $$\sum n^nx^n \notin \mathbb C\{x\}$$