Questions tagged [topological-rings]
127 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
- 50,341
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
- 9,162
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 square…
Jacques
- 643
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
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…
A Rock and a Hard Place
- 1,490
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 $I$-adic…
Riley
- 81
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
- 3,865
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
- 8,318
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
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
- 4,150
- 16
- 26
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
- 1,671
6
votes
1 answer
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
- 61