Mathematics

Questions tagged [topological-rings]

126 questions
37
votes
2 answers

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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15
votes
2 answers

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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11
votes
0 answers

A natural topology on a field

I can endow any field with a natural topology in the following way. Given a polynomial $f\in K[X]$ , I denote by $\mathcal{O}(f)=\{x\in K\mid\exists y\in K^{\times}\ f(x)=y^2\}$ , i.e. the set of elements $x\in K$ such that $f(x)$ is a non-zero…
Jacques
  • 735
10
votes
1 answer

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
  • 2,110
9
votes
1 answer

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,…
xyzw14
  • 93
8
votes
1 answer

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 E Hanson
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7
votes
1 answer

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 needs…
A Rock and a Hard Place
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7
votes
2 answers

Convergence of the sequence $nq^n$ for a topologically nilpotent element $q$ of a topological ring $R$.

Let $R$ be a topological ring and $q$ an element of $R$ such that \begin{align*} \lim_{n\to \infty}q^n = 0, \end{align*} then we call $q$ topologically nilpotent. Do we also have \begin{align*}\lim_{n\to\infty}nq^n=0?\end{align*} If $q$ is…
user1347143
  • 118
7
votes
0 answers

Matsumura's Commutative Ring Theory, Theorem 8.14

I read Matsumura's Commutative Ring Theory . I cannot understand the proof of Theorem 8.14 (2) $\Longrightarrow$ (3) on page 62. Here is the statement. Theorem 8.14. Let $A$ be a Noetherian ring and $I$ an ideal. If we consider $A$ with the $I$…
Riley
  • 91
7
votes
1 answer

Sufficient condition for multiplication to be continuous

Given a commutative ring $(R,+,\cdot)$ , and a topology $\tau$ on $R$ such that for any $a\in R$ the maps \begin{align} \cdot a: &R\to R \\ &x\mapsto a\cdot x \\\\ &\mbox{and} \\\\ +a: &R\to R \\ &x\mapsto x+a \end{align} Are continuous, is it…
Carlyle
  • 4,217
7
votes
1 answer

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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7
votes
1 answer

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
7
votes
2 answers

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
  • 188
6
votes
1 answer

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
  • 5,243
6
votes
1 answer

Any Cauchy sequence on a topological vector space is bounded

Let $(X, \tau)$ be a topological vector space. We say that a set $E \subset X$ is bounded if, for all neighborhood $V$ of $0$ , there exists $r > 0$ such that $E \subset rV$ . Now let $\{x_n\}$ be a Cauchy sequence on $X$ , that is, given an…
ThiagoGM
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