Questions tagged [topological-rings]

This is intended to include topological fields as well.

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3 votes
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Initial topology for a topological ring

Given a topological ring $R$ and an arbitrary (thus not necessarily surjective) epimorphism $q: R \to S$ of underlying rings is there a finest topology on $S$ such that 1) $S$ is a topological ring ...
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7 votes
1 answer
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Regular epimorphisms in the category of topological rings

I am cross-posting this question from math stack exchange (link below) as it has not received any comments or answers over the past month. A regular epimorphism is a morphism $f: X \to Y$ that is the ...
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2 votes
0 answers
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References on topological ringed spaces

This is a follow up to this question of mine. First of all, let me fix some terminologies, which may or may not be standard: Definition: A topological ringed space is a pair $X := (|X|, \mathcal{O}_X)...
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1 vote
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204 views

Automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers?

I am interested to see what is currently known about the automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers (with respect to the $p$-adic topology induced by the $p$-...
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1 vote
1 answer
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S-unital compact rings are profinite

It is well-known that compact Hausdorff topological unital rings are profinite. The proof generalises to (left or right) s-unital rings (i.e. rings such that for all $r\in R$ we have $r\in Rr$ or for ...
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4 votes
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Associativity equation for topological rings and logarithms

Let $R$ be a topological ring of characteristic zero. Assume that $R$ is commutative. Let $G :R \times R \to R$ be a continuous function satisfying $G(G(x,y),z)=G(x,G(y,z))$ and $G(0,x)=x$, for all $x,...
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  • 423
1 vote
0 answers
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Properties of differentiable functions on non-locally-bounded fields

I was reading some results on the structure of non-locally-bounded topological fields, and I was wondering what is known about differentiable functions on them. In particular, on the complex numbers ...
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2 votes
1 answer
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Krull dimension of completions in non-noetherian setting (especially completed perfections)

What are some general results describing the Krull dimension of the completion of a non-Noetherian ring with a "nice" topology? An example of the sort of "nice" topological ring I'm looking for is a ...
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1 vote
0 answers
104 views

Completed Tensorproduct

I am trying to understand the completed tensorproduct. This can be defined as follows: Given a topological ring $R$ and two linearly topologized rings $A$ and $B$ with fundamental systems of open ...
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6 votes
0 answers
137 views

m-systems and n-systems in topological rings

Note that throughout rings have a multiplicative identity and are not necessarily commutative Definition: Let $R$ be a ring and let $M\subseteq R$. Then, $M$ is an m-system iff for every $x,y\in ...
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9 votes
1 answer
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Is $k(\!(x,y)\!)$ a topological field?

More generally, let $(R,m)$ be a Noetherian local domain with fraction field $K$. The $m$-adic topology turns $R$ into a topological ring. When $R$ is a discrete valuation ring, this topology extends ...
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