All Questions
Tagged with or ra.rings-and-algebras ra.rings-and-algebras
1,143 questions with no upvoted or accepted answers
55
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answers
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What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?
Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the ...
- 1,238
38
votes
0
answers
1k
views
Groups whose complex irreducible representations are finite dimensional
By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...
- 38.6k
30
votes
0
answers
1k
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Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?
Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields?
I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
- 1,742
30
votes
0
answers
3k
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Greatly expanded new edition of a Bourbaki chapter on algebra?
Recently I discovered by accident that Bourbaki issued in 2012 a radically expanded version of their 1958 Chapter 8 Modules et anneaux semi-simples (like other chapters, initially in French) within ...
- 52.9k
28
votes
0
answers
527
views
What algebraic structure characterizes all natural operations between differential operators and differential forms?
On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms:
the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$;
...
- 37.8k
23
votes
0
answers
463
views
Topological loops vs. algebro-geometric suspension in Hochschild homology
Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...
- 6,069
17
votes
0
answers
1k
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Relations in a certain Lie algebra
Let ${\mathfrak g}$ be the (real) Lie algebra generated by infinitely many generators $D_i, E_i$ with $i=1,2,3,\dots$ subject to the following relations for any natural numbers $i,j$:
\begin{gather*}
[...
- 114k
17
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0
answers
704
views
When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
- 7,300
16
votes
0
answers
574
views
Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector-valued cross product exists?
It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real ...
- 1,206
16
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0
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784
views
How to explain the picturesque patterns in François Brunault's matrix?
How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...
- 19.6k
16
votes
0
answers
860
views
Is "being a full ring of quotients" a Morita invariant property?
Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
- 2,454
15
votes
0
answers
217
views
If a map between unital rings preserves multiplication and successor, does it preserve addition?
Welcome to my first MathOverflow posting!
This is a question about rings, all of them assumed to be both unital and associative.
Let $f\colon R\to S$ be a map between rings such that $f(xy)=f(x)f(y)$ ...
- 151
14
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0
answers
557
views
Rings that fail to satisfy the strong rank condition
In T.Y. Lam's book Lectures on Modules and Rings, a ring $R$ is said to satisfy the strong rank condition if, for every natural number $n$, there is no right $R$-module monomorphism $R^{n+1}\to R^n$. ...
- 282
14
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0
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378
views
A hard Lefschetz theorem for nilCoxeter algebras
Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter
algebra (over the reals, say), as defined at
https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has
a natural ...
- 50.8k
14
votes
0
answers
1k
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Kaplansky's theorem and Axiom of choice
Kaplansky in his paper titled by Projective Modules gave an important and essential theorem as follow:
Theorem: Let $R$ be a ring, $M$ an $R$-module which is a direct sum of (any number of) countably ...
- 1,788
14
votes
1
answer
2k
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Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
- 2,550
13
votes
0
answers
573
views
Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
- 2,339
13
votes
0
answers
571
views
Why is it so hard to give examples of differentially closed fields?
The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
- 32.4k
13
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0
answers
355
views
Analog of Haar element in an algebra
In a Hopf algebra $H $ (over some field $ k $), there is the notion of a Haar element $ h \in H$. This is an element of the algebra which has the property that if $ V $ is a representation of $ H $, ...
- 4,612
13
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0
answers
338
views
When does Hochschild homology commute with infinite products?
Let $A$ be an associative algebra. Its zeroth Hochschild homology $\mathrm{HH}_0(A)$ is the cokernel of the linear map $A^{\wedge 2} \to A$, $a \wedge b \mapsto ab - ba$. I.e. you quotient the ...
- 54.6k
12
votes
0
answers
542
views
Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
- 63.1k
12
votes
0
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224
views
Do compact inverse-property loops (or just compact Moufang loops) have invariant uniformities and bi-invariant Haar measure?
So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...
- 2,585
12
votes
0
answers
185
views
Hopf-Galois extensions where the "extension" is a module?
For $H$ a Hopf-algebra, an $H$-Hopf-Galois extension is a map of rings $\phi\colon\thinspace A\to B$ such that $H$ coacts on $B$ over $A$, $B\otimes_AB\cong B\otimes H$, and the cofixed points, or the ...
- 10.4k
12
votes
0
answers
267
views
Finitely generated skew-fields
There is a well known theorem saying that a commutative field that is finitely generated as a ring has to be finite (Kaplansky).
Is the same true for non-commutative "fields" (usually called ...
12
votes
0
answers
276
views
How can you unitalize a higher category?
Given an associative nonunital algebra $A$, there are (at least) two standard ways to produce a unital algebra $A'$ together with a map $A \to A'$. Following the discussion in the comments below, ...
- 54.6k
12
votes
0
answers
533
views
Does there exist a Noetherian ring of finite injective dimension but higher Krull dimension?
Definition: a (not necessarily commutative) left and right Noetherian ring $R$ is said to be Auslander-Gorenstein if
(i) $R$ has finite left and right injective dimension (in which case it turns out ...
- 221
12
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0
answers
605
views
Given an algebra, can it be realized as a block of a Hopf algebra?
During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in k$...
- 2,450
11
votes
0
answers
436
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
11
votes
0
answers
450
views
A congruence involving roots of unity
Let $f(x) \in \mathbb{Z}[x]$ and suppose $f(\omega^j) \in \mathbb{Z}$ for all $j= 1, \dots, n$ where $\omega = e^{2 \pi i/n}$ is a primitive $n^{\text{th}}$ root of unity.
Computational evidence ...
- 305
11
votes
0
answers
411
views
Lazard's theorem and Hopf structures on the polynomial algebra
Let $k$ be an algebraically closed field of characteristic $0$. A well-known result of Lazard's states that an algebraic group which is isomorphic as a variety to an affine space is unipotent (M. ...
- 426
11
votes
0
answers
841
views
Is "Determinant" a Hochschild coboundary?
Assume that $n>2$.
Is there an associative unital algebra structure on $\mathbb{C}^{n}$ such that $D$, the determinant as a $n-\text{form} $ on $\mathbb{C}^{n}$,
would be a Hochschild ...
- 356
11
votes
0
answers
400
views
Detecting a module for the free group algebra on a finite quotient
Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module
$M = R / (ax + by + c) R$.
I am ...
- 19.2k
11
votes
0
answers
305
views
Detecting invertible elements in group rings by their images for finite quotients of the group
Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent).
Consider an element ...
11
votes
0
answers
708
views
What happens to simple modules under Ringel duality?
If $A$ is a quasi-hereditary algebra then its Ringel dual $A'=End_A(T)$ is the endomorphism algebra of a (minimal) full tilting module $T$ for $A$. The algebra $A'$ is again quasi-hereditary, although ...
- 578
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0
answers
286
views
What do Multilinear Forms tell us about Representations?
The last few days I have been calculating whether certain group representations are real, complex, or quaternionic. It is well-known that the type of the representation corresponds to what type of ...
- 5,191
10
votes
0
answers
488
views
Reconstruction of commutative differential graded algebras
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.
Here, differentials ...
- 889
10
votes
0
answers
317
views
Near-ring spaces
$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
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10
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446
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In search of lost graded rings
$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
- 11.8k
10
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0
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474
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 ...
- 3,785
10
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0
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227
views
What is the meaning of the coefficients of the Alekseev-Torossian associator
Drinfeld associators became a central object in mathematics and mathematical physics. They appear in deformation quantization, quantum groups, in the proof of the formality of the little disks operad, ...
- 8,385
10
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219
views
Recover the field from its Milnor K-groups
For every field $F$, consider $K_n^M(F)$ the $n$-th Milnor K-group of $F$ for each $n \in \Bbb N$, and form the Milnor K-ring $K^M(F)=\oplus_{n \geq 0}K^M_n(F)$. For instance, $K_1(F)=F^{\times}$.
...
- 6,592
10
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0
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416
views
Equational theory in the signature (+,*,0,1) of sedenions and beyond
Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, ...
- 2,013
10
votes
0
answers
237
views
Generalized eigen property of a matrix
Given a $n \times n$ invertible matrix $A$, I am interested in the set
$$
\mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}.
$$
Thus, for all eigenvalues $\lambda_i$, we have $...
- 909
10
votes
0
answers
241
views
Has anyone seen this construction of dg algebras?
Let $A$ be an associative algebra, $M$ a right $A$-module. Suppose we are given an $A$-module homomorphism $M \to A$. Then we can make $M$ itself into an associative algebra via the multiplication
$$ ...
- 40.2k
10
votes
0
answers
854
views
Scholze's infinite to finite type ring theory reductions?
In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking.
The most virtuosic pages in Scholze's ...
- 119
10
votes
0
answers
275
views
Are local finite dimensional Hopf algebras symmetric?
Recall that a finite dimensional algebra $A$ over a field $K$is called a Frobenius algebra in case $A \cong D(A)$ as right modules, where $D(A) \cong Hom_K(A,K)$. In case $A \cong D(A)$ as bimodules, ...
- 26.5k
10
votes
0
answers
341
views
Which endomorphism algebras are not Morita-trivial?
Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that:
$\mathcal C$ is ...
- 54.6k
9
votes
0
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460
views
Does the book "Algebra III" exist (within the Encyclopaedia of Mathematical Sciences series from Springer)?
Within the series "Encyclopaedia of Mathematical Sciences", as published by Springer, one finds the 8 volumes, namely,
the volumes I, II, IV, V, VI, VII, VIII, IX but zbMath has no listing ...
- 161
9
votes
0
answers
365
views
A characterisation of symmetric algebras using Hochschild (co)homology
A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for ...
- 26.5k
9
votes
0
answers
213
views
What are the algebras for the codensity monad of $\textbf{FP-Ring} \rightarrow \textbf{Ring}$
Let $\textbf{Fin-Set}$ denote the category of finite sets, and let $\textbf{Set}$ denote the category of sets. The inclusion functor $\textbf{Fin-Set} \rightarrow \textbf{Set}$ from the category of ...
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