**16**

votes

**1**answer

617 views

### Does GL_n(Z) have a noetherian group ring?

Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL_n(Z)$ been considered in the literature?
Motivation: a recent trend has been to study ...

**3**

votes

**1**answer

61 views

### Does every equivalence class in a Brauer-Wall group have a (graded) division algebra?

It is known that each equivalence class in a Brauer group has a division algebra (or, in other words, every central simple algebra is isomorphic to $\mathrm{Mat}(D)$ for some division algebra $D$). Is ...

**15**

votes

**0**answers

273 views

### Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google:
Suppose I have a countable field, $k$. ...

**1**

vote

**0**answers

63 views

### Example H-unital algebra which is not unital

What is an example of an algebra which is H-unital (that is its Bar complex is acyclic) and yet it is not unital?

**4**

votes

**2**answers

330 views

### A question about pairs of lines in 3D projective space

Consider a 3-dimensional projective space $X$.
Let $m$ be the smallest number so that there are $m$ pairs of lines
$ \ell_1,\ell'_1$, $ \ell_2,\ell_2'$, ... , $\ell_m, \ell'_m$ in $X$:
a) For ...

**4**

votes

**2**answers

467 views

### Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v2 §19)

$\newcommand{\refone}{\textbf{(1)}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which is ...

**2**

votes

**0**answers

49 views

### Augmentation ideal of the cohomology of an elemntary abelian 2-group [closed]

Let V be an elemntary abelian 2-group and $R=H^{*}V$ its cohomology.
What is the Augmentation ideal of R and what is the quotient of R by its augmentation ideal ?

**0**

votes

**0**answers

171 views

### A noncommutative analogy of the tube lemma

Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to ...

**1**

vote

**0**answers

41 views

### Valuations in Higher-dimensional local fields

I have the following question which I believ should be true but I would like to have a different opinion about it:
Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and ...

**1**

vote

**0**answers

133 views

### Reductive Group Actions and Completion

Suppose that $A$ is a Noetherian (not necessarily commutative) $\mathbb{C}[h]$-algebra equipped with a rational action of an affine reductive group $G$, i.e., $A$ is a $\mathbb{C}[G]$-comodule and ...

**7**

votes

**1**answer

221 views

### Why can't one modify Kaplansky's proof to conclude that every projective module is a direct sum of its finitely generated projetive submodules?

Due to the examples given in the answer to this question, I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in this paper and replacing every occurrence ...

**3**

votes

**1**answer

184 views

### Possibilities for dimensions of $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ for a local ring

Let $R$ be a local commutative ring with maximal ideal $\mathfrak{m}$, and denote by $k$ the residue field $R/\mathfrak{m}$. Then we can look at the sequence of $k$-vectorspaces
$$R/\mathfrak{m}, ...

**1**

vote

**1**answer

140 views

### Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?

**1**

vote

**2**answers

193 views

### Is there an intuitionistic generalized boolean algebra (of Stone)?

A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...

**2**

votes

**3**answers

136 views

### Does a BCL algebra define a partial order?

A BCL algebra is a universal algebra with a binary operation denoted as "$*$" and a $0$-ary operation (constant) denoted as "$0$", satisying the following axioms:
(1) $x * x = 0$;
(2) if $x * y = 0$ ...

**6**

votes

**1**answer

100 views

### Infinite dimensional simple algebras of finite degree

Let $F$ be a field and let $B$ be an $F$-algebra. The degree of $B$ over $F$ is the smallest positive integer $\deg_F B = d \geq 1$ such that every element of $B$ satisfies a (monic) polynomial of ...

**1**

vote

**0**answers

46 views

### On generating Euler Square of index q, q-1 (where q is any prime power)

Can somebody help me in generating Euler Square of index q, q-1 (where q is any prime power). Also, kidly tell me if there is any code availeble for generating the stated Euler Square. The details of ...

**5**

votes

**0**answers

336 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

**1**answer

205 views

### Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a ...

**11**

votes

**7**answers

735 views

### Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...

**4**

votes

**0**answers

123 views

### Formal DG-algebras

Sorry for this question but I really have difficulties with model categories.
Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...

**3**

votes

**0**answers

66 views

### Quasi-isomorphisms and Subalgebras

Let $A$ and $B$ $dg$-algebras over $\mathbb{C}$. If there exists an isomorphism $f:A\to B$, then every subalgebra $A'$ of $A$ is isomorphic to the subalgebra $f(A')$ of $B$.
What is if $f$ is ...

**15**

votes

**5**answers

2k views

### Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...

**2**

votes

**1**answer

206 views

### Jacobson radical and group rings/subalgebras

Let $G$ be a finite group and $N\le G$ be a subgroup. Consider the group algebra $kN$ as a subalgebra of $kG$ over an algebraically closed field $k$ of positive characteristic.
What can we deduce ...

**4**

votes

**1**answer

191 views

### coherent modules

Let $R$ be a nontrivial ring. A right $R$-module $M$ is called coherent if ${\rm Ker} (f)$ is finitely generated for any $R$-module homomorphism $f: L\to M$ with $L$ finitely generated.
It is ...

**1**

vote

**0**answers

78 views

### Units in residue classes

Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field)
Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units ...

**7**

votes

**0**answers

226 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 ...

**6**

votes

**0**answers

148 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 ...

**0**

votes

**0**answers

86 views

### Relationships between finiteness of stable rank and IBN property of rings

Does any ring of finite stable rank have IBN property? Where can we find this result?

**3**

votes

**0**answers

115 views

### Isomorphic bound quiver algebras for different admissible Ideals

We know that for a path algebra KQ, whether or not KQ is finite dimensional (namely, Q may or may not have oriented cycles), there might be different admissible ideals I and J of KQ for which the ...

**4**

votes

**1**answer

367 views

### Re-interpreting vector spaces in a choice-less model of ZF as modules over a regular ring in ZFC

I am searching a module $M$ over a (von Neumann) regular ring $A$ ($\forall a\in A$, $\exists x\in A$: $axa=a$) with two properties:
(1) every finitely generated submodule of $M$ is projective ...

**1**

vote

**1**answer

202 views

### Existence of a left adjoint to tensor product implies projectivity

Let $S$ and $R$ be two (not necessarily commutative) $k$-algebras for $k$ a field. If I have a $S$-$R$ bimodule $_SM_R$, I can form the functor $_SM_R\otimes_R (-):R\text{Mod} \rightarrow ...

**0**

votes

**1**answer

81 views

### Norm of a number in an algebraically closed field

I consider an algebraically closed field of characteristic zero $F$ as a vector space over a real closed field $R \subset F$. I would like to define a norm on $F$ in an invariant fashion, i.e. if we ...

**5**

votes

**1**answer

250 views

### A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...

**1**

vote

**1**answer

152 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 ...

**2**

votes

**1**answer

251 views

### Trace of finitely generated projective module

Let $k$ be a field and $A$ a $k$-algebra with unit. The trace module is
$$
T(A)=A/[A,A],
$$
where $[A,A]$ is the left $A$-module generated by all elements of the form $ab-ba$ for $a,b\in A$. The ...

**1**

vote

**0**answers

79 views

### Simplest (?) example of bicrossed product Hopf algebra

Suppose we have two Hopf algebras, H and A and additionally A is (left) H-module algebra and H is (right) A comodule coalgebra. This means that A is left module over H and moreover that
...

**3**

votes

**2**answers

316 views

### Are algebraic structures uniquely identifed by their free objects?

It might be a naive question, as I am not a specialist in this field.
This is a follow-up to this question.
I want to study varieties of objects generalizing ordered monoids, in particular using an ...

**3**

votes

**2**answers

459 views

### Examples of complete distributive lattices that are not Heyting algebras

Here is a short question with a possibly simple and short answer:
I need an example of a complete distributive lattice that is not a Heyting algebra which should be an infinite complete lattice that ...

**0**

votes

**0**answers

68 views

### approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...

**4**

votes

**1**answer

160 views

### Potentially identity elements in an Abelian group

I didn't see this problem before. I motivated by the questions
Is every commutative group structure underlying at least one (unitary, commutative) ring structure
A basic question about rings
...

**1**

vote

**0**answers

114 views

### Non commutative analogy of compact-open topology

Let $R$ be a ring, define a topology on $AUT(R)$(Or End(R)) with the following subbase:
For every two 2-sided Ideal $I$ and $J$, a subbase element is $B(I,J)=\{f\in AUT(R) \mid f(I)+J=R\}$.
We can ...

**1**

vote

**1**answer

244 views

### A question in ring theory

Is there an example of two groups $G_{1}, G_{2}$ such that there are
two non isomorphic ring $R_{1}$ and $R_{2}$ such that the additive group of both rings is isomorphic to $G_{1}$ and their unit ...

**1**

vote

**2**answers

310 views

### A basic question about rings

Perhaps this is a trivial question, but I have no idea how to justify it.
Call a pair of groups $(G_1, G_2)$ ring-compatible if $G_1$ is abelian and there exists a ring $R$ with addition and ...

**3**

votes

**2**answers

178 views

### Examples of cancellative normal semigroups

I've got a couple of things to test against normality in cancellative semigroups. A normal semigroup $S$ is one in which for any $x\in S$ we have $xS=Sx.$ This implies the Ore condition $$x,y\in ...

**1**

vote

**0**answers

116 views

### subring of the matrix algebra

Let $Mat_2(\mathbb{Z})$ be the $\mathbb{Z}$-algebra of $2\times2$ matrices with integer entries.
Let $A$ be a $\mathbb{Z}$-submodule of $Mat_2(\mathbb{Z})$ containing $\mathbb{Z}$. We want to show ...

**3**

votes

**1**answer

206 views

### On rings $R$ for which $R \cong \frac RI$ for any proper two-sided ideal $I$

This is a problem I asked in SE, but it seems the question is more suitable for MO.
Consider a ring $R$ (not necessary with identity or commutative) such that for any proper two-sided ideal $I$, ...

**2**

votes

**0**answers

56 views

### Cyclic modules over serial rings

Let $R$ be a serial ring. What can be said about the uniform dimension of cyclic $R$-modules? Specially I would like to know Is it true that every cyclic $R$-module has finite uniform dimension ?

**4**

votes

**0**answers

100 views

### Sum of projective submodules of a projective over a semihereditary ring

Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a ...

**2**

votes

**2**answers

202 views

### How much information does the multiplicative semigroup of an algebra contain?

How much do we know about an given algebra when we only know its semigroup strucure under the product law?
How far can two algebras be distinguished by knowing only their semigroup strucure?
The ...