**21**

votes

**0**answers

2k views

### Sums of injective modules, products of projective modules?

Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?
Analogously, under what assumptions on R does ...

**13**

votes

**0**answers

595 views

### Countable Hom/Ext implies finitely generated

Today I learned this interesting fact from Jerry Kaminker: If $A$ is an abelian group such that $\mathrm{Hom}(A,\mathbb{Z})$ and $\mathrm{Ext}(A,\mathbb{Z})$ are both countably generated, then in fact ...

**12**

votes

**0**answers

543 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 ...

**11**

votes

**0**answers

134 views

### Must nonunit in group algebra of free group generate proper two-sided ideal?

Let $F$ be a free group and $k$ be a field. If $x$ is an element of
the group algebra $k[F]$ that is not a unit (equivalently, that is not
a nonzero scalar multiple of an element of $F$), must the ...

**11**

votes

**0**answers

514 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 ...

**10**

votes

**0**answers

516 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 ...

**9**

votes

**0**answers

98 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 ...

**9**

votes

**0**answers

125 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 ...

**9**

votes

**0**answers

289 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 ...

**9**

votes

**0**answers

229 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 ...

**8**

votes

**0**answers

162 views

### What's the analogue of a Young symmetrizer in the Brauer algebra?

According to Schur--Weyl duality, the centralizer of $\mathrm{GL}(V)$ acting diagonally on $V^{\otimes N}$ is the group algebra of the symmetric group $\mathbb S_N$. An equivalent formulation is the ...

**8**

votes

**0**answers

107 views

### Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...

**8**

votes

**0**answers

185 views

### Does this kind of non-noetherian bimodule exist?

Question: Do there exist simple rings $R$ and $S$ (i.e., rings with no proper nonzero ideals) and an $(R,S)$-bimodule $M$ such that
$M$ is finitely generated both as a left $R$-module and a right
...

**8**

votes

**0**answers

159 views

### Depth for non-commutative rings

The depth of a ring or module is one of the most basic invariants in commutative ring theory.
Q1: Is there also a powerful notion of depth for non-commutative rings ?
By a search in mathscinet, I ...

**8**

votes

**0**answers

364 views

### Is the class of additive groups of rings axiomatizable?

I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...

**8**

votes

**0**answers

462 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 ...

**7**

votes

**0**answers

238 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 ...

**7**

votes

**0**answers

134 views

### Is 2 a zerodivisor in the ring parametrizing rank-n algebras?

Let $n$ be a natural number; I am studying (commutative) rings $R$ and $R$-algebras $A$ such that $A\cong R^n$ as $R$-modules. There is a universal such algebra: a ring $R_0$ and a free, rank-$n$ ...

**7**

votes

**0**answers

544 views

### duality between universal enveloping and function algebra for GL(n)

Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular ...

**6**

votes

**0**answers

240 views

### Can every commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

All rings here are associative, commutative and unital. By a ring of characteristic zero (resp. of characteristic $p$, for prime $p$) I mean a ring $A$ such that the canonical homomorphism $\mathbb ...

**6**

votes

**0**answers

147 views

### How to prove that a projective module is not free?

Let $A$ be a noncommutative (perhaps $\ast$-) algebra (over $\mathbb{C}$) and let $M$ be a projective module defined via a projector $P\in M_n(A)$; i.e. $M=P(A^n)$. Furthermore, assume that all ...

**6**

votes

**0**answers

111 views

### flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...

**6**

votes

**0**answers

153 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 ...

**6**

votes

**0**answers

167 views

### Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?

**6**

votes

**0**answers

153 views

### Series in topological rings that only converge if almost all summands are zero

While trying to understand a certain topological ring better, I stumbled onto the following question.
Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...

**6**

votes

**0**answers

161 views

### Is it possible to define Hopf algebra as an object in the category of (associative) algebras?

I think this must be well-known, but I can't find references, so my apologies from the very beginning.
Consider first the notion of bialgebra. It is usually defined as an object $B$ in the category ...

**6**

votes

**0**answers

137 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 ...

**6**

votes

**0**answers

396 views

### Ideal class group isomorphic to $\mathbb{Z}$

Hi everybody,
I wonder if someone could provide me with a simple example of a Dedekind ring whose ideal class group is isomorphic to $\mathbb{Z}$.
The point is I would like the example simple enough ...

**6**

votes

**0**answers

279 views

### Is there an idempotent measure on the free LD system?

This is a follow up question to MO question "Idempotent measures on the free binary system?".
Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law:
...

**6**

votes

**0**answers

394 views

### Can we write unitary matrices as positive linear combinations of Hermitian matrices?

The space $M_n:=M_n(\mathbb{C})$ of complex $n\times n$ matrices has the structure of a finite-dimensional complex vector space.
The space of Hermitian matrices forms a cone in this vector space ...

**6**

votes

**0**answers

343 views

### Quantum polynomial rings and singularities

Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...

**5**

votes

**0**answers

87 views

### presentations of subalgebras

Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x_1,...,x_n$ subject to finitely many ...

**5**

votes

**0**answers

103 views

### Noncommutative group of invertible ideals of a ring

Let $R$ be a noetherian domain and let $\mathcal{O}$ be an $R$-algebra that is finitely generated and projective as an $R$-module. The set of invertible fractional ideals of $\mathcal{O}$ is a group ...

**5**

votes

**0**answers

109 views

### Centralizers of elements in free group algebras

Let $A$ be a group algebra of a free group, and $x \in A$. What is the centralizer of $x$? Is there something like Bergman's theorem for free associative algebras?

**5**

votes

**0**answers

249 views

### A generalization of real characters on a group

Yesterday I understood that I can't live without this construction:
Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps ...

**5**

votes

**0**answers

165 views

### Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients.
$$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$
Monomials $x^k$ are mapped to $n ...

**5**

votes

**0**answers

135 views

### Software for BMW algebra calculations?

Does software exist for computations in the BMW algebra?
For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...

**5**

votes

**0**answers

338 views

### The logarithm over $\mathbb F_1$

In 'Cyclotomy and analytic geometry over F1', Manin proposes a version of the notion of `analytic function' over the 'field with one element $\mathbb F_1$'.
Question 1: can somebody explain or give ...

**5**

votes

**0**answers

203 views

### On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists ...

**5**

votes

**0**answers

222 views

### How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...

**5**

votes

**0**answers

274 views

### Homological dimension of completed path algebras.

Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations.
Is it true that the I-adic completion of A has finite homological dimension?

**5**

votes

**0**answers

214 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 ...

**5**

votes

**0**answers

362 views

### Generators of associated graded algebra

Suppose that $A = \bigcup_{n=0}^{\infty} A_n$ is a filtered algebra over a field $k$. The associated graded algebra is $\mathrm{gr} A = \bigoplus_{n=0}^{\infty} A_n/A_{n-1}$, where we define $A_{-1} ...

**5**

votes

**0**answers

410 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 ...

**5**

votes

**0**answers

100 views

### Does Mittag-Lefflerness descend?

I have read in the Stacks Project that if $A \to B$ is a faithfully flat ring homomorphism, $M$ is an $A$-module, and $M \otimes_A B$ is a flat, Mittag-Leffler $B$-module, then $M$ is a flat, ...

**5**

votes

**0**answers

144 views

### integral versus adjoint action on Hopf algebra

Suppose that $H$ is a finite dimensional Hopf algebra (with counit $\varepsilon$) and $T$ is a non zero right integral of $H^{\star}$ (the dual Hopf algebra). Let $ad_h$ be the adjoint action on $H$, ...

**5**

votes

**0**answers

245 views

### Testing isomorphism of finitely generated algebras

Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and
$C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated ...

**5**

votes

**0**answers

224 views

### Roots of unity in algebraic K-theory

For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit.
For ...

**5**

votes

**0**answers

89 views

### Ring properties which can be checked on sufficiently many Ore localizations

Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of ...

**5**

votes

**0**answers

325 views

### Lie locally nilpotent associative algebras

Let $A$ be an associative algebra over a field. Then $A$ can be regarded as a Lie algebra via the Lie bracket defined by $[a,b]=ab-ba$ for every $a,b\in A$.
The algebra $A$ is called Lie locally ...