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Results tagged with ra.rings-and-algebras
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user 36688
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
3
votes
1
answer
228
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An isomorphic invariant in ring theory
Let $R$ be a unital ring. We define the Murray Von Neumann relation $M$ on $R$ as follows:
We say $a M b$ iff $a=xy,\;b=yx$ for some $x,y\in R$. (This is inspired by the usual Murray Von Neumann eq …
- 356
4
votes
2
answers
567
views
Some examples of clean topological spaces
I asked this question at MSE but I did not received any answer, so I repeat it here at MO:
What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X) …
- 356
3
votes
2
answers
164
views
Is the pair $(C([0 \;1]),\mathbb{C})$ a consecutive pair?
Motivated by this post we give the following definition:
Definition: A pair of unital rings $(R,S)$ is called a consecutive pair of rings if they do not have non trivial idempotent and …
- 356
1
vote
1
answer
303
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A question in ring theory
Is there an example of two groups $G_{1}, G_{2}$ such that there are
two non isomorphic ring $R_{1}$ and $R_{2}$ such that the additive group of both rings is isomorphic to $G_{1}$ and their unit gro …
- 356
1
vote
0
answers
104
views
Hochschild coboundary on the space of alternative forms
Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is
an element $\phi \in C^{k}(A)$ …
- 356
1
vote
0
answers
134
views
Non commutative analogy of compact-open topology
Let $R$ be a ring, define a topology on $AUT(R)$(Or End(R)) with the following subbase:
For every two 2-sided Ideal $I$ and $J$, a subbase element is $B(I,J)=\{f\in AUT(R) \mid f(I)+J=R\}$.
We can re …
- 356
1
vote
1
answer
304
views
Simple $C^*$ algebras whose all commutator elements have scalar square
Is there a simple $C^*$ algebra $A$, not isomorphic to $M_2(\mathbb{C})$, such that for every commutator element $x=ab-ba$, $x^2$ is an scalar element?
- 356
2
votes
0
answers
301
views
A question on Giles Gardam counter example to the Unit conjecture of Kaplansky
The unit version of the Kaplansky conjecture is about units in $FG$ where $F$ is a field and $G$ is a torsion free group. In a recent counter example by Giles Gardam, it is given an e …
- 356
4
votes
2
answers
659
views
Modules over infinite rings which can not be a finite union of their proper submodules
It is well known that a vector space over an infinite field cannot be a finite union of its proper subspaces.
Does this fact have an immediate and obvious generalization to modules over infinite di …
- 356
7
votes
0
answers
637
views
Row rank and column rank of matrix with entries in a commutative ring
Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed
in "Ranks of Modules"
one can say that the row rank of $A$ is …
- 356
0
votes
1
answer
185
views
A subset (or subgroup) associated to a group
Edit: According to comment conversations we revise the question.
Let $G$ be a group. We consider the following subset of $G$:
$$\{g\in G \mid e^{\lambda_g} \in \mathbb{C}\lambda (G)\},$$
where $\lamb …
- 356
1
vote
1
answer
185
views
Is a finite dimensional graded algebra isomorphic to the equivariant de Rham complex of a Li...
Edit: According to essential comment of YCore I revise the question.
Let $A$ be a finite dimensional graded algebra which is a unital, super commutative and associative algebra. Is there a Lie group …
- 356
1
vote
0
answers
177
views
A locally convex $C^*$ algebra without zero divisor
Let we have a locally convex $C^*$ algebra $A$. That is $A$ is a TVS equipped with an algebra and an involution structure such that all operations are continuous. Moreover the topology on $A$ i …
- 356
2
votes
0
answers
88
views
Non-existence of idempotent via evaluation of higher order cocycle on a tuple of idempotents
The Kaplansky conjecture and Kaplanski-Kadison conjecture are classical conjectures about non existence of non-trivial idempotents in group algebra $\mathbb{C}\Gamma $ or the reduced group algebra …
- 356
1
vote
0
answers
112
views
Idempotent conjecture and non-abelian solenoid
Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation c …
- 356