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.

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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 ...
Dmitri Zaitsev's user avatar
37 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 ...
Benjamin Steinberg's user avatar
31 votes
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3k views

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 ...
Jim Humphreys's user avatar
29 votes
0 answers
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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(...
Watson's user avatar
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28 votes
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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 $M$; ...
Dmitri Pavlov's user avatar
23 votes
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455 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 ...
Aaron Mazel-Gee's user avatar
17 votes
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1k views

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*} [...
Terry Tao's user avatar
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17 votes
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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 (...
Libli's user avatar
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16 votes
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552 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 ...
pregunton's user avatar
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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 ...
Stefan Kohl's user avatar
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16 votes
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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 ...
Torsten Schoeneberg's user avatar
14 votes
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498 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 $R^{n+1}\to R^n$. ...
Karl Lorensen's user avatar
14 votes
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1k views

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 ...
Ali Reza's user avatar
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1 answer
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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 ...
Adam Epstein's user avatar
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13 votes
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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 $, ...
Joel Kamnitzer's user avatar
13 votes
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361 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 ...
Richard Stanley's user avatar
13 votes
0 answers
319 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 ...
Theo Johnson-Freyd's user avatar
12 votes
0 answers
471 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; ...
Gro-Tsen's user avatar
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12 votes
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Do compact inverse-property loops (or just compact Moufang loops) have 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 ...
Harry Altman's user avatar
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12 votes
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181 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 ...
Jonathan Beardsley's user avatar
12 votes
0 answers
257 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 ...
Miroslav Korbelar's user avatar
12 votes
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274 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, ...
Theo Johnson-Freyd's user avatar
12 votes
0 answers
520 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 ...
Christopher's user avatar
12 votes
0 answers
603 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 $q\in k$...
Julian Kuelshammer's user avatar
11 votes
0 answers
433 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 ...
David Roberson's user avatar
11 votes
0 answers
438 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 ...
user94267's user avatar
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11 votes
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838 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 ...
Ali Taghavi's user avatar
11 votes
0 answers
396 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 ...
JSE's user avatar
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11 votes
0 answers
291 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 ...
Jean-Claude Sikorav's user avatar
11 votes
0 answers
679 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 ...
Andrew's user avatar
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11 votes
0 answers
282 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 ...
ARupinski's user avatar
  • 5,181
10 votes
0 answers
474 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 ...
Walterfield's user avatar
10 votes
0 answers
308 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}\...
Emily's user avatar
  • 10k
10 votes
0 answers
427 views

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}\...
Emily's user avatar
  • 10k
10 votes
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464 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 ...
Drew Heard's user avatar
  • 3,655
10 votes
0 answers
218 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, ...
DamienC's user avatar
  • 8,093
10 votes
0 answers
206 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, $K_1(F)=F^{\times}$. ...
sawdada's user avatar
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10 votes
0 answers
398 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, ...
user107952's user avatar
  • 2,063
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 $...
Jiro's user avatar
  • 909
10 votes
0 answers
239 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 $$ ...
Dan Petersen's user avatar
  • 39.1k
10 votes
0 answers
836 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 ...
Stan's user avatar
  • 119
10 votes
0 answers
391 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. ...
Paul Gilmartin's user avatar
10 votes
0 answers
258 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, ...
Mare's user avatar
  • 25.7k
10 votes
0 answers
336 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 ...
Theo Johnson-Freyd's user avatar
9 votes
0 answers
405 views

Does Wedderburn's Theorem hold constructively?

Wedderburn's 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 of ...
Martin Brandenburg's user avatar
9 votes
0 answers
350 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 ...
Mare's user avatar
  • 25.7k
9 votes
0 answers
208 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 ...
Ronald J. Zallman's user avatar
9 votes
0 answers
352 views

Duality between coalgebras and (pseudocompact) algebras - uniqueness

The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below. Let $k$ be a field. The ...
Julian Kuelshammer's user avatar
9 votes
0 answers
139 views

Where is it shown that a countable self-injective ring is semilocal?

In Lawrence, John. "A countable self-injective ring is quasi-Frobenius." Proceedings of the American Mathematical Society (1977): 217-220. the first line is this: ...
rschwieb's user avatar
  • 1,593
9 votes
0 answers
259 views

Name for rings with $R \cong R^{\mathrm{op}}$

Is there a name for rings that are isomorphic to their opposite ring $R^{\mathrm{op}}$ in the literature? I'm especially interested for the class of Artin algebras.
Mare's user avatar
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