Questions tagged [topological-rings]
104 questions
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
- 8,289
9
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
- 1,960
1
vote
1 answer
Is closure of an ideal also an ideal?
In a topological ring $R$ (assume identity if necessary), is it true that if $I$ is an ideal then $\bar{I}$ , i.e. closure of $I$ is also an ideal?
It's easy to show that if $x, y \in \bar{I}$ then so is $-x$ and to show $x+y$ lies in closure, we…
Bhaskar Vashishth
- 11,279
1
vote
1 answer
Example of Banach ring with non-closed ideal
I was wondering if there are commutative Banach rings $(A, \|\!\cdot\!\|)$ with unit which contain an ideal $\mathfrak a \subseteq A$ such that $\mathfrak a$ is not closed w.r.t. to the topology induced by the norm $\|\!\cdot\!\|$ on $A$. I haven't…
johnnycrab
- 1,952
- 12
- 14
0
votes
2 answers
How can we add small numbers to an element of a topological ring to make neighborhoods?
In this paper by Kaplansky on topological rings, in lemma 7 (reading the proof is not important to understanding my doubt), right ideal $B$ is given dense in $eA$ where $A$ is a topological ring (Hausdorff) and $e$ is an idempotent. Being dense, it…
blabla
- 1,104