All Questions
Tagged with ra.rings-and-algebras reference-request
329 questions
0
votes
2
answers
288
views
Equivalent Forms of AC
There are many algebraic equivalences of AC in the literature. A famous one is "every ring with identity has a maximal ideal".
Where can I find this equivalences, specially those in rings theory !? ...
- 521
3
votes
0
answers
167
views
A non-commutative ring from SU(2)
$SU(2)$, which will be regarded here as the group of unit quaternions under multiplication, has 3 conjugacy classes of finite subgroups which don't have cyclic subgroups of index 1 or 2. They are:
...
- 3,645
3
votes
0
answers
155
views
Clifford algebra is graded separable
Let $D$ be an algebra of odd differential operators on a free module $V$, this algebra is isomorphic to the Clifford algebra $Cl(V^* \oplus V)$. Let $m$ denote multiplication map $$m : D\otimes D \to ...
- 1,545
13
votes
3
answers
357
views
How should one look at the set of compatible ring structures on a given group?
Earlier today I had a conversation with a friend about ways of putting topologies on sets of first-order structures; we wound up talking about reducts and expansions from a topological point of view, ...
CommunityBot
- 1
5
votes
0
answers
442
views
A reference on semisimple linear algebra
Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...
CommunityBot
- 1
5
votes
0
answers
272
views
Cocontinuous monadic functors
Some forgetful functors of algebraic categories preserve colimits, but most do not. In order to understand this phenomen in general, I have classified cocontinuous monadic functors whose target is an ...
- 63.1k
4
votes
1
answer
686
views
Character theory of $2$-Frobenius groups.
This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/...
CommunityBot
- 1
33
votes
2
answers
2k
views
What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
- 6,282
3
votes
1
answer
213
views
What is this deformed group algebra named?
I have a semisimple algebra $R$ over a field $k$ that looks like a group algebra $k[G]$ except that it's deformed slightly. That is, instead of a basis $e_1 ..., e_n$ closed under multiplication, I ...
CommunityBot
- 1
6
votes
0
answers
168
views
Classifying algebras with two idempotent generators and involution
Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$.
For example,...
- 1,021
2
votes
1
answer
209
views
Simplicial complex made of central idempotents of an algebra
Let $A$ be an algebra, say over $\mathbb{C}$ and finite-dimensional, but not necessary semisimple. I have the strong feeling, which I would like to prove and use, about the following rather natural ...
- 38.6k
11
votes
2
answers
2k
views
Generalizing the Fundamental Theorem of Symmetric Polynomials
The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...
1
vote
1
answer
560
views
A good introduction to S unit equations
I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper.
http://faculty.nps.edu/pstanica/research/...
- 30.5k
16
votes
1
answer
1k
views
Lagrange's theorem for Hopf algebras
Under what conditions is a Hopf algebra free over any of its sub-Hopf algebras?
I am reading "Hopf algebras and their actions on rings" by Susan Montgomery, specifically chapter 3. Lagrange's theorem ...
- 3,078
0
votes
1
answer
465
views
Weak algebraic structures
The following question can be thought as a sequel of this one.
Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...
CommunityBot
- 1
6
votes
2
answers
832
views
Constructing a ring from an abelian group in a minimal way
I'm looking for a specific construction, taking an abelian group (with designated element) $(G,+,1)$ to a commutative ring $(R,+,\cdot,1)$, where $G\subset R$ as a pointed abelian group, and which is ...
- 1,979
15
votes
1
answer
1k
views
Are wild problems related to undecidable ones?
In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, here). In logic, there is a notion of an ...
CommunityBot
- 1
3
votes
2
answers
590
views
Algebra with positive definite symmetrizing trace is semisimple.
This is a follow-up question to
When does a symmetric algebra over a field of characteristic 0 fail to be semisimple?
Let $H$ be a symmetric algebra over $\mathbb{R}$ with symmetrizing trace $\tau:...
CommunityBot
- 1
10
votes
1
answer
1k
views
Is there a way to define a prime ideal object via diagrams in the category of rings?
I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. ...
CommunityBot
- 1
27
votes
1
answer
2k
views
Strong group ring isomorphisms
Background/Motivation
Let $R$ be a commutative ring with unit. If $G$ is a finite (or in general, discrete) group, let $R[G]$ be the group $R$-algebra associated to $G$. The isomorphism problem for ...
- 23k
4
votes
1
answer
497
views
Is the functor of divided powers a weakly monoidal functor?
Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every $m\...
- 22.6k
5
votes
1
answer
618
views
Commutator formulas in a universal enveloping algebra
I am interested in finding formulas for commutators of symmetrized monomials in a universal enveloping algebra. Let $C(x_1,\ldots, x_n)= (1/n!)\sum x_{\sigma(1)}\cdots x_{\sigma(n)}$ where the sum ...
- 8,098
1
vote
0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922
7
votes
1
answer
2k
views
"Linear algebra" over Z/nZ - reference please!
Let A be a matrix with entries in Z/nZ. (n is not assumed to be prime.) Then the size of the row span is the size of the column span. All computations are mod n, so both these numbers are finite.
I ...
- 25.8k
3
votes
0
answers
123
views
Name/references for analogue of ring with 2-cocycle condition instead of distributivity
I'm looking for a name for, and any past study on, the following kind of algebraic structure:
A set S equipped with an additive operation making it an abelian group, and a multiplication $*:S \times ...
- 7,320
4
votes
1
answer
305
views
Doing Real Algebraic Geometry on *-Rings
I've been searching google and scholar google, but i only have come upon orderings and Hermitian forms on *-fields.
Has real algebraic geometry been carried over to *-rings? *-rings are rings with an ...
- 25.5k
6
votes
2
answers
2k
views
Ideals in a noncommutative ring such that their product is their intersection?
If $R$ is a commutative ring and $I$ and $J$ are ideals in $R$ such that $I+J=R$ then $I \cap J=IJ$. This is not generally true in noncommutative rings, e.g. let $R$ be the lower triangular 2 x 2 ...
- 2,392
8
votes
0
answers
521
views
Skew polynomial algebra
When I was a very little hare, a big grey wolf told me about the following skew polynomial algebra, which I never understood. My question is whether the following construction is a part of some bigger ...
- 12.3k
15
votes
1
answer
633
views
Introduction to "commutative semialgebra"?
Of course, commutative algebra is a fundamental topic in algebraic geometry, number theory, representation theory, and so on.
However, there are some instances (most obviously tropical geometry) ...
- 85.6k