# Questions tagged [topological-rings]

86 questions

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### Haar Measure of a Topological Ring

A topological ring is a (not necessarily unital) ring $(R,+,\cdot)$ equipped with a topology $\mathcal{T}$ such that, with respect to $\mathcal{T}$, both $(R,+)$ is a topological group and $\cdot:R\times R\to R$ is a continuous map. A left Haar…

Batominovski

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### group of units in a topological ring

I am looking at some notes on Adeles and Ideles by Pete Clark here, and puzzling over exercise 6.9 (page 6), that if the group of units $U$ in a topological ring is an open subset, then multiplicative inversion on $U$ is continuous. I am supposing…

user43208

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### How to recover the topology of a topological ring using Yoneda lemma

Consider the category of topological rings. By the Yoneda embedding, suppose $A$ is a topological ring, if the functor $\mathrm{Hom}(-,A)$ is given, then we can recover the topological ring $A$ from this functor $\mathrm{Hom}(-,A)$. My question is,…

xyzw

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### Connected field must be path-connected?

A topological ring is a ring $R$ which is also a Hausdorff space such that both the addition and the multiplication are continuous as maps.
$F$ is a topological field, if $F$ is a topological ring, and the inversion operation is continuous, when…

David Chan

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### Is the ring $R$ a topological ring with respect to the following topology?

Background
This question is motivated by trying to answer this question. But before going into the question straight let me give some background.
Definition 1. Let $R$ be a ring and $A$ be an ideal of $R$. Let $\mathcal{T}_A$ denote the set of all…

user170039

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### Quotients of topological rings

Let $\varphi\colon R\to S$ be a surjective ring homomorphism and let $R$ be a topological ring.
Is there some nice characterization of the finest topology on $S$ for with both $S$ becomes a topological ring and $\varphi$ becomes continuous?
If…

A Rock and a Hard Place

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### Unconditional convergence of a sum of elements in a complete Hausdorff topological ring.

I'm not that familiar with theoretical math in general (I studied engineering), but I recently ended up down a theoretical rabbit hole that led me to the following question:
Is there some type of well known property (e.g. locally compact, locally…

dch

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### Is there an infinite topological meadow with non-trivial topology?

For reference meadows are a generalization of fields that were designed to be compatible with the requirements of universal algebra. Specifically a meadow is a commutative ring equiped with an involution $x\mapsto x^{-1}$ which obeys
$$x\cdot x\cdot…

James Hanson

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### Group Rings of Topological Groups and Fields

Suppose $\Bbb{K}$ is a topological field and $G$ is a topological group. Recall that $\Bbb{K}[G]$ denotes the group ring of $G$ over $\Bbb{K}$, which consists of sums of the form $\sum_{g \in G} a_g g$ with at most finitely many of the $a_g \in…

user193319

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### Making the subgroup of units of a topological ring a topological group

I'm a beginner in the theory of topological algebra (I'm reading something about it in Robert's "A course in $p$- adic analysis").
At page 24, the author states that if $A$ is a topological ring, the subgroup $A^{\times}$ is not in general a…

LBJFS

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### Locally compact valued field is complete?

In J. P. Serre's book Local Fields, in the proof of one proposition it says:
If $K$ is locally compact, it is complete.
How did he deduce that? As far as I know, there is no correlation between locally compactness and completion. So I assume he used…

BlueDiamond

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### Counterexamples to the Artin-Rees Lemma

This well known Lemma about $I$-stable filtrations asserts:
Lemma (Artin-Rees) Let $A$ be a Noetherian ring and $E$ a finitely generated $A$-module.
Let $F$ be a submodule of $E$ and $\{E_i\}$ an $I$-stable filtration. Then the induced filtration on…

Sabino Di Trani

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### Extension to an automorphism of topological fields

Here is what I am trying to prove:
Let $L$ be an algebraically closed topological field of characteristic zero and $K$ be a proper algebraically closed and topologically closed subfield of $L$. Let $\lambda,\mu\in L\setminus K$ and…

user829347

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### Is $\mathbb R$ with usual euclidean topology, homeomorphic with some topological field of positive characteristic?

Does there exists a topological field of positive characteristic which is homeomorphic with $\mathbb R$ with the usual topology ?
By homeomorphism here, I mean just topological homeomorphism, not necessarily preserving any algebraic structure.

user521337

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### $\mathbb{Z}_p[\mathbb{Z}/p^{n}\mathbb{Z}]\cong \mathbb{Z}_p[T]/\left((T+1)^{p^n}-1\right)$ as topological rings?

Consider the group-ring $\mathbb{Z}_p[\mathbb{Z}/p^{n}\mathbb{Z}]$ with the product topology, and the quotient ring $\mathbb{Z}_p[T]/((1+T)^{p^n}-1)$ with the quotient topology, ($\mathbb{Z}_p[T]$ has the $(p,T)$-adic topology).
If $\gamma$ is a…

JoseCruz

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