# Questions tagged [topological-rings]

This is intended to include topological fields as well.

16
questions

3
votes

1
answer

61
views

### Topology of the Malcev-Neumann group ring

For a ring $R$ and a group $G$ the group ring $R[G]$ consist of maps from $G$ to $R$ with finite support.
It was shown that if the group is fully ordered them this ring can be embedded in a division ...

2
votes

0
answers

43
views

### Topological rings with a final topology

Given a family of ring homomorphisms $ \phi_i : X \rightarrow Y_i $ where each $ Y_i $ is a topological ring and consider the initial topology on $ X $, i.e. the coarest topology such that each map is ...

3
votes

1
answer

127
views

### Ideals of an ordered ring

Suppose $R$ is a strictly ordered (non-commutative) ring, in particular $ab > 0$ for any $a,\, b > 0$, that is also discrete in that there are no elements between $0$ and $1$.
Now consider a two-...

2
votes

0
answers

130
views

### Embedding a monoid into a group via its monoid ring

Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...

2
votes

0
answers

285
views

### What is the status of topological problems in group rings?

I have extensively studied group rings and structure of their units and also zero divisors and normalizer problem in integral group rings.. I was pondering upon questions like what if we use ...

3
votes

0
answers

114
views

### 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 ...

7
votes

1
answer

222
views

### 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 ...

2
votes

0
answers

74
views

### 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)...

1
vote

0
answers

250
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 $p$-...

1
vote

1
answer

45
views

### 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 ...

4
votes

0
answers

99
views

### 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,...

1
vote

0
answers

23
views

### 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 ...

2
votes

1
answer

396
views

### 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 ...

1
vote

0
answers

113
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 ...

6
votes

0
answers

142
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 ...

9
votes

1
answer

754
views

### 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 ...