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I came up with this:

Let $S=\{(a_n)_{n\geq 1}\:|\:a_n\in \mathbb{C}, (a_n)_{n\geq 1}\text{ converges}\}$ be the set of convergent complex sequences. Then this set forms a ring under pointwise operations, with the multiplicative identity given by the constant sequence $1,1,\dots$, and the additive identity given by $0,0,\dots$.

Let $S_0=\{(a_n)_{n\geq 1}\in S\:|\:\text{for sufficiently large }n, a_n=0\}$ be the set of sequences that are "eventually $0$". Then $S_0$ is an ideal in $S$.

Define $\overline{S}=S/S_0$, and for each sequence $s\in S$, write $\overline{s}=s+S_0\in\overline{S}$. Then two sequences $s_1,s_2\in S$ are equal in $\overline{S}$ if they are "eventually identical".

Define $L:\overline{S}\rightarrow\mathbb{C}$ by $L(\overline{s})=\lim_{n\rightarrow\infty}s$. This map is well-defined, because if $\overline{s_1}=\overline{s_2}$, then $s_1$ and $s_2$ are "eventually" the same, so they have the same limit. By the familiar limit rules, $L$ is a ring homomorphism that maps unity to unity.

Let $I\subseteq\overline{S}$ be the kernel of this map. Of course, $I$ is a proper ideal.

Let $s\in S$ be a sequence that does not converge to $0$, so that means $\overline{s}\in\overline{S}\setminus I$. Then for all sufficiently large $n$, $s_n\neq 0$. So it is possible to form a sequence $s'$ such that for all sufficiently large $n$, $s'_n=1/s_n$. This means that the sequence $ss'$ is "eventually" $1$, and thus $\overline{s}$ is a unit in $\overline{S}$.

What I have just shown is that every element in $\overline{S}\setminus I$ is a unit, and since $I$ is a proper ideal, every element in $I$ is not a unit. Thus, $I$ is the unique maximal ideal in $\overline{S}$, which means that $\overline{S}$ is actually a local ring, which is nice.

Is there a name for this construction? I don't think it tells us anything new about convergent sequences of complex numbers, it's just a nice way to "repackage" what we know about convergent sequences.

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The construction of this local ring is an instance of the following more general observation:

Let $X$ be a topological space and $p \in X$. Then the ring of germs of continuous functions at $p$ is local with maximal ideal $\{f : f(p)=0\}$.

Indeed, a convergent sequence (say, with values in $\mathbb{C}$) is the same as a continuous map $$\mathbb{N} \cup \{\infty\} \to \mathbb{C},$$ where $\infty$ is mapped to the limit of the sequence. Here, $\mathbb{N} \cup \{\infty\}$ has the topology in which $\mathbb{N}$ is discrete and the sets $\mathbb{N}_{\geq n} \cup \{\infty\}$ form a basis of the neighborhoods of $\infty$. Thus, continuity is precisely the limit condition. In other words, $\mathbb{N} \cup \{\infty\}$ is the one-point compactification of the discrete space $\mathbb{N}$. A germ of a continuous function at $\infty$ is therefore a convergent sequence (defined on some $\mathbb{N}_{\geq n} \cup \{\infty\}$, but we can extend randomly on $\mathbb{N} \cup \{\infty\}$), where two such sequences are identified when they are eventually the same. The observation tells us that these form a local ring, and that the maximal ideal consists of the sequences that converge to $0$.

So, the answer to your question "Does this local ring have a name?" is: Yes, it is the ring of germs of continous functions on the one-point compactification of $\mathbb{N}$ at $\infty$.

The mentioned general observation precisely says that the stalk of the sheaf of continuous functions on a space at every point is a local ring, that is, $(X,C(-))$ is a locally ringed space.

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  • $\begingroup$ Thanks for the answer! I'm sorry to say I don't know any of those words, so forgive me for asking this slightly rude question: is there any use in describing complex sequences in this way -- does it make it easier to prove certain facts about them -- or is this just a nice way to "package" what we already know about complex sequences? More rudely: is this just sophisticated verbiage, or is it good for something? $\endgroup$ Dec 3, 2023 at 4:24
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    $\begingroup$ No worries. That's exactly why I linked the Wikipedia articles where one can learn more about all these notions. About your follow-up question: I am not sure. The facts that you are repackaging are useful for sure, and it is always good to have the more general picture in mind. But I would be surprised if you can prove anything about convergent sequences with this interpretation which you cannot prove directly. For me, the point of my answer is that it is no coincidence that you get a local ring like you described. It is part of a more general lemma about germs of continuous functions. $\endgroup$ Dec 3, 2023 at 4:36
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    $\begingroup$ I have edited my answer to make it more approachable. $\endgroup$ Dec 3, 2023 at 13:04
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    $\begingroup$ Thank you! I understand better now. $\endgroup$ Dec 4, 2023 at 2:22

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