Questions tagged [ra.rings-and-algebras]
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.
3,366
questions
2
votes
1
answer
124
views
Non-counital coalgebras
For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
- 7,658
24
votes
14
answers
4k
views
Ring with three binary operations
A rather precocious student studying abstract algebra with me asked the following question: Are there interesting rings where there are not just two but three binary operations along with some ...
- 263
7
votes
1
answer
462
views
Characterisation of finite dimensional C*-algebras?
$\DeclareMathOperator\Spec{Spec}$Let $A$ be a finite dimensional $*$-algebra over $\mathbb C$.
(Namely, an associate algebra equipped with an involution $*:A\to A$ satisfying $(ab)^*=b^*a^*$ and $(\...
- 2,233
4
votes
0
answers
175
views
Question about basis of $\text{Der}_{k}(k[X])$
Let $k[X] = k[x_1,\ldots, x_n]$ be the polynomial ring over a field of characteristic zero.
Assume that $(D_1,\ldots, D_n)$ is a $k[X]$-basis of $\text{Der}_k(k[X])$. Suppose that the vector space $\...
- 291
12
votes
3
answers
812
views
Axiomatic definition of quantum groups
This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, axiomatically, a quantum group?
There are ...
7
votes
2
answers
673
views
Ostensibly different products on Ext-groups
The following is presumably not the greatest generality in which this question makes sense.
Given a ring $k$, graded-commutative if it helps, and a Hopf-algebra $A$ over $k$, there is a Yoneda ...
- 3,486
12
votes
1
answer
573
views
Unit group of octonions over finite fields
One can define the algebra $A(K)$ of octonions over an arbitrary field $K$, see for example the command OctaveAlgebra in GAP: https://www.gap-system.org/Manuals/doc/ref/chap62.html .
When $K$ is a ...
- 85.1k
2
votes
1
answer
261
views
For two polynomials, what is the relationship between the least linear combination and the resultants?
Let $k$ be a field and $k[x]$ be the ring of polynomials over $k$. Given two polynomis $m_1(x), m_2(x) \in k[x]$, I want to know the relationship between the resultants and the least linear ...
- 21.8k
5
votes
1
answer
708
views
Rigid monoidal and closed monoidal categories
I am trying to understand the relationship between rigid monoidal categories and closed monoidal
categories. First every rigid monoidal category is closed, with an adjoint to the functor $X \otimes -$ ...
- 29.8k
3
votes
1
answer
514
views
Lagrange’s interpolation formula: Theoreme and Example [closed]
I would like to know where they come up with the formula of Lagrange interpolation (Lagrange’s interpolation formula),Lagrange_polynomial because I did some research, but I find a different definition ...
- 30k
3
votes
0
answers
146
views
Units in group rings
Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...
- 60.4k
2
votes
2
answers
588
views
Good reference on the algebraic geometry of non-associative rings
I am looking for a good reference about the algebraic geometry of non-associative rings. I am in particular interested in derivation algebras.
Preferrably an online resource or a book that is ...
- 11.4k
3
votes
0
answers
75
views
On noncommutative transcendence degrees
The original transcendence degree for (noncommutative) division algebras is the Gelfand-Kirillov transcendence degree, due to I. Gelfand and K. Kirillov ([ Sur les corps li´es aux algèbres ...
- 2,733
7
votes
1
answer
324
views
Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group
My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. ...
- 7,658
3
votes
1
answer
2k
views
Prime ideals of formal power series ring that are above the same prime ideal
Let $R$ denote a commutative ring with identity and let $R[[X]]$ denote the
ring of formal power series over $R$ in an indeterminate $X$. If $I$ is an ideal of $R$,
then $I[[X]]$, the set of power ...
- 938
9
votes
1
answer
669
views
Curious anti-commutative ring
Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context?
Here $\Lambda[x_0, x_1, x_2, \ldots]$ is the (graded-)commutative algebra (either over ...
- 3,486
-2
votes
3
answers
777
views
Oldest abstract algebra book with exercises?
Per the title, what are some of the oldest abstract algebra books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of ...
- 178k
2
votes
0
answers
54
views
Gelfand-Kirillov dimension for non-associative algebras
Let $A$ be any finitely generated algebra - non necessarely unital neither associative - over a base field $k$. Let us denote the product $*$. Suppose $A$ is finitely generated by $S$, and introduce $...
- 2,733
4
votes
0
answers
302
views
Polynomial objects in any concrete category
EDIT: The original question had a trivial answer: it's just a coproduct. New question below
New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...
- 430
3
votes
0
answers
156
views
A characterization for a commutative ring with a special intersection property for prime ideals
Let $R$ be a commutative ring with $1$ with the property that for any infinite family $\{P_i\}_{i\in I}$ of distinct prime ideals of $R$ we have $\cap_{i\not= j} P_i\subseteq P_j$ for all but fnitely ...
- 61
3
votes
2
answers
248
views
Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial
Let $G$ be a finite group. Let $p$ be a prime.
Let $O_p(G)$ be the $p$-core of $G$.
Are there any theorems known saying something like
$O_p(G)$ is trivial, if and only if ... and
$O_p(G)$ is non-...
- 5,282
3
votes
0
answers
212
views
On the Gelfand-Kirillov Conjecture
The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...
- 2,733
8
votes
0
answers
117
views
Literature and history for: lifting matrix units modulo various kinds of ideal
This is not so much a mathematics question as a cross between a "history of mathematics" question and a reference request.
My PhD student has been working on some problems concerning ...
- 42.8k
2
votes
1
answer
260
views
Universal constructions that factor through endomorphisms
If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor $...
- 859
0
votes
3
answers
945
views
non-associative but commutative algebra [closed]
Is it possible(or may be easier) to give an example of non associative algebra but commutative?
At first sight, it seems possible to prove associativity from commutativity but later realised it may no ...
- 72.3k
6
votes
0
answers
292
views
Independence results on pure algebra
I think that the most celebrated result in this direction is Shelah's famous work on Whitehead's Problem:
Is every abelian group $A$ such that $Ext^1(A, \mathbb{Z})=0$ free?
This is known to be ...
- 2,733
3
votes
1
answer
179
views
Gelfand-Kirillov dimension of the first Weyl algebra
How can we compute the Gelfand-Kirillov dimension (GK for short) of the first Weyl algebra?
As we know we can look at the Weyl algebra as a generalized Weyl algebra in the following way:
Let $A=\...
- 98
10
votes
0
answers
465
views
Is $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p$ coherent?
The question is as in the title:
Is the ring $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p = \mathbb{Z}_p \otimes_{\mathbb{Z}_{(p)}} \mathbb{Z}_p$ coherent?
As shown in the related question, the ...
- 2,956
2
votes
0
answers
141
views
Characterization of algebraic integers providing a prime ideal
Let $\alpha$ be an algebraic integer and let $\mathcal{O}_{\mathbb{Q}(\alpha)}$ be the ring of integers of $\mathbb{Q}(\alpha)$.
Question: How to characterize an algebraic integer $\alpha$ such that $\...
- 25.9k
13
votes
1
answer
776
views
Is $\mathbb{Q}_p \otimes_{\mathbb{Q}}\mathbb{Q}_p $ coherent?
Let $\mathbb{Q}_p$ denote the field of fractions of $\mathbb{Z}_p$. By the answers to this quesition the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}_p$ cannot be a Noetherian ring (...
- 27.3k
1
vote
0
answers
154
views
When every localization of the polynomial ring over a ring has finitely many idempotents
Let $R$ be a commutative ring such that every localization ring $R_r$ has finitely many idempotents for each non nilpotent element $r\in R$. Why dose every localization ring $R[x]_{f(x)}$ have ...
- 11
3
votes
1
answer
152
views
Behavior of invariants under reduction mod p
Let $R$ be a finitely generated $\mathbb{Z}$-algebra with an [edit: linear algebraic] action of $G(\mathbb{Z})$ where $G$ is a split simply-connected semisimple group.
Then for any prime $p$ we have a ...
- 3,474
8
votes
1
answer
485
views
Category of modules over an Azumaya algebra and the Brauer group
Let $k$ be a field, and let $\alpha \in \mathrm{Br}(k)$. Let $A$ be an Azumaya algebra representing $\alpha$. Then the category $A$–$\mathrm{mod}$ depends only on $\alpha$.
I would like to know ...
- 115k
1
vote
0
answers
104
views
Field theory, Abel-Ruffini theorem, technical question
Let me put the question first.
Let $F,K$ be subfields of $\mathbb{C}$. Suppose that $t,\rho\in \mathbb{C}$ are algebraic over $F$ and $\rho \in K$. If $F(t)\cap K\subset F$, is it true that $F(t,\rho)\...
- 11
5
votes
2
answers
433
views
Exact subcategory with trivial Grothendieck group: what are the consequences and examples
Let $C$ be (a full) exact subcategory of the category of $R$-modules. We suppose that $C$ is essentially small. If the Grothendieck group $K_{0}(C)=0$, what can be said about the higher groups $K_{n}(...
- 101
4
votes
1
answer
448
views
Exterior algebra of normed spaces
This question is related to my prior question, but this one is aimed, even though it's more general. If $V$ is a vector space, we define the exterior algebra of $V$ do be:
$$\bigwedge V := \bigoplus_{...
- 42.2k
2
votes
0
answers
169
views
if $I$ is finitely presented nilpotent and $M/IM$ is finitely presented, then $M$ is finitely presented
Let $R$ be a commutative ring, and let $I \subseteq R$ be a nilpotent ideal. Let moreover $M$ be an $R$-module, and let $IM$ be the submodule generated by the products $xm$ with $x \in I$ and $m \in M$...
- 2,069
1
vote
1
answer
44
views
Contraction elements in unital *-rings
Let $A$ be a unital *-ring. Let us put, $A^+=\{\sum_1^N x^*_ix_i: x_i\in A , N\in \mathbb{N} \}$. We write $x\geq0$ if $x\in A^+$.
Suppose that for every $n\in \mathbb{N}$ and $a\in A$, there exsits $...
- 3,992
31
votes
1
answer
2k
views
Is this formal noncommutative power series identity known?
I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series
$$ 1 + \...
- 16.2k
2
votes
3
answers
310
views
Efficient algorithm for matrix equation $AXB + BXA = F$
For $n\in\mathbb{N}$, let $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times n}$ be two symmetric positive definite matrices and let $F\in\mathbb{R}^{n\times n}$ be arbitrary.
Is there any ...
- 19.4k
3
votes
1
answer
91
views
Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks
A version of Brauer's second main theorem is as follows:
Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
- 42.8k
5
votes
0
answers
73
views
Reference request: associative subalgebras of Cayley algebras are at most 4-dimensional
By a Cayley algebra I mean an 8-dimensional algebra (over an arbitrary field) formed in the Cayley-Dickson process. (They are also called octonion algebras, but I prefer to reserve the term octonion ...
- 22.9k
6
votes
1
answer
358
views
Example of a projective module with non-superfluous radical
Let $R$ be a ring with unit. A submodule $N$ of an $R$-module $M$ is called superfluous if the only sumbodule $T$ of $M$ for which $N+T = M$ is $M$ itself.
It is shown, for example, in
[1] F. W....
- 315
1
vote
1
answer
203
views
Second summand to make projective module free
Suppose there's a projective $R$-module $P$ (non-free). We know that there is another $R$-module $M$ such that $P\oplus M$ is free over $R$. Is there a way to write down such an $M$ in terms of $P$?
...
- 22.5k
4
votes
3
answers
334
views
Coinvariants of tensor products of Hopf algebras
Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way.
The axioms of Hopf algebras imply that
$$
G^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...
- 1,008
6
votes
0
answers
332
views
Independence of characters with respect to polynomials
I came across the following property :
Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors,
$\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\mathcal{...
- 4,618
5
votes
2
answers
335
views
Gelfand-Kirillov dimension of generalized Weyl algebras
I believe that the Gelfand-Kirillov (GK) dimension for a generalized Weyl algebra $D(\sigma,a)$ is just the GKdim$(D) + 1$.
Does anyone have a reference for this?
I can find partial results, and I ...
- 11.4k
3
votes
1
answer
366
views
Smallest ring whose field of fractions includes all the reals (subring of omnific integers?)
The surreal numbers have a subring, the ring of "omnific integers" or $\mathbf{Oz}$, which have the property that every surreal number is a quotient of two omnific integers. That is, the ...
- 2,382
12
votes
1
answer
375
views
Characteristic polynomial of an $8 \times 8$ symmetric matrix with indeterminate entries related to octonionic multiplication
I consider $1,i,j,k,l,m,n,o$ the standard basis of the (complexified if you like) octonions ($\mathbb{O}$ for the octonions).
Let $a = x_1.1 +\ldots + x_8.o$, $b = x_9.1+ \ldots + x_{16}.o$ and $c = ...
- 106k
4
votes
0
answers
121
views
Is the average associator over a finite subloop of octonions necessarily zero?
For any three octonions $a,b,c$, their associator is defined as \begin{equation*} [a,b,c]=a(bc)-(ab)c \end{equation*} and measures their non-associativity so to speak.
Now suppose that $L$ is a finite ...
- 439