**0**

votes

**1**answer

73 views

### Reference request for stably free modules

I am looking for some references on the theory of stably free modules. I will call (F) the following property for a ring $R$: every f.g. stably free module over $R$ is free.
1) Is there a standard ...

**6**

votes

**1**answer

155 views

### Is the following module over a group ring necessarily infinitely generated?

Suppose $\Gamma$ is a (finitely presented, but this is probably irrelevant) group, and $M$ is a finitely generated (EDIT: finitely presented) module over $\mathbb{Q}\Gamma$ which is ...

**2**

votes

**1**answer

101 views

### A quadratic algebra with four generators and four relations

Algebra that I'm going to describe pop-up in my research, it looks completely elementary, but I don't know any appropriate references.
Let $k$ be an algebraically closed field of characteristic ...

**1**

vote

**3**answers

360 views

### smooth connected affine scheme over Z has good reduction almost everywhere

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a ...

**9**

votes

**0**answers

140 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 ...

**12**

votes

**1**answer

353 views

### Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...

**3**

votes

**2**answers

178 views

### Finite Dimensional Simple nonunital associative Algebras

I have the following problem:
Let K be any field. An finite dimensional associative non-unital algebra A is a vector space A, togeter with a K-biliniear associative operation such that there is NO ...

**5**

votes

**0**answers

251 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 ...

**1**

vote

**0**answers

106 views

### Examples of Lie subalgebras of universal enveloping algebras

I'm looking for non-trivial examples of triples $(\mathfrak g, L, \psi)$, where $\mathfrak g$ and $L$ are finite-dimensional non-abelian Lie algebras over field $\Bbbk$ and $\psi\colon U_{\mathfrak ...

**1**

vote

**0**answers

78 views

### Examples of noncommutative Bezout domains

I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...

**2**

votes

**1**answer

94 views

### Near-ring localizations

Are there any known results on localization for near-rings (i.e., "rings" with non-abelian addition and only one-sided distributive law)? The books on near-rings I checked don't mention this topic at ...

**10**

votes

**1**answer

422 views

### Can a division algebra have degree divisible by its characteristic?

I apologize in advance if this is easy, but I've tried Googling, and had no luck.
I'm currently working on a proof, and I realized in the course of writing that this proof will break if out there ...

**2**

votes

**0**answers

73 views

### Orders of certain quotients of power series rings

Let $\Lambda_d := \mathbb{Z}_p[[T_1, \ldots, T_d]]$ denote the ring of formal power series in $d$ variables over the ring of $p$-adic integers. Suppose that $g \in \Lambda_d$ is an irreducible ...

**2**

votes

**3**answers

593 views

### Is there a general notion of semigroup action?

The analogous result to Cayley’s theorem or Yoneda lemma in semigroup theory represents semigroups as semigroups of functions from a set to itself. This suggests that a semigroup action consists of a ...

**15**

votes

**1**answer

415 views

### If $B\subseteq A$ are free & finite rank $R$-algebras, is $R\to A \otimes_B R$ injective?

(In this question, all rings and algebras are commutative with identity.)
I have a situation that boils down to the following data: a ring $R$, an $R$-algebra $A$ with a subalgebra $B$ such that $A$ ...

**5**

votes

**0**answers

171 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 ...

**2**

votes

**0**answers

102 views

### Separating duality for TVS?

What is the modern concept (term) for "separating duality" (dualité séparante in french) in the sense of Bourbaki (TVS Ch II § 6) as explained in the following ?
...

**0**

votes

**1**answer

98 views

### $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$

In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals),
an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$.
[See Comm. Rings by Kaplansky, ...

**4**

votes

**1**answer

403 views

### Who defined and who coined “module”?

The title of my Q. says it all:
QUESTION: Who defined and who coined: module?
Would it be Emmy Noether?
EDIT In view of @anon's and KConrad's answers, and as it could have been ...

**2**

votes

**0**answers

220 views

### Kaplansky Idempotent conjecture and Extension theory

We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...

**3**

votes

**1**answer

189 views

### Linear dependency of real numbers with integer coefficients adding up to zero [closed]

Let $x = (x_1, \dots, x_n)$ be a vector of real number. I was asking myself if there was an efficient way of telling whether there exists a non-zero vector of integers $z \in \mathbb Z$ such that both
...

**0**

votes

**0**answers

111 views

### A special Lie subalgebra

Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra ...

**6**

votes

**2**answers

250 views

### String diagrams for bimodules over noncommutative algebras?

I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal ...

**2**

votes

**0**answers

56 views

### Gelfand Kirillov dimension of induced modules

How does the Gelfand-Kirillov dimension of induced modules behave?
In patricular, if $S$ is an Ore extension of $R$, how is the GK-dimension of an $R$-module say $N$ related to the GK dimension of $N ...

**3**

votes

**1**answer

217 views

### What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...

**5**

votes

**1**answer

314 views

### Noncommutative HKR theorem

What is the analog of HKR theorem in the noncommutative world?
Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a smooth commutative algebra $A$ of finite type ...

**4**

votes

**1**answer

178 views

### Isomorphism of matrix ring over ore domain

Let $R_1,R_2$ be (left and right) ore domains. Does $ Mat_n(R_1)\cong Mat_m(R_2)$ implie m=n and $q.f.(R_1)\cong q.f.(R_2)$?
An counter example, a proof or a reference is welcomed.
Thanks

**2**

votes

**1**answer

150 views

### General criterion to find a Z-basis in a fixed generating subset

Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$
be a fixed finite subset.
Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the ...

**9**

votes

**1**answer

408 views

### Motivation behind the definition of hochschild cohomology

For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...

**12**

votes

**3**answers

438 views

### Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$?
To prevent things from being too easy, I ...

**3**

votes

**2**answers

204 views

### Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...

**2**

votes

**1**answer

117 views

### Projecting solutions of Hermitian forms over local rings

Let $R$ be a local ring (commutative and with $1$) with maximal ideal $M$, with an involution $\theta$. Let $h$ be a Hermitian form on $R^n$, i.e. $h:R^n\times R^n\rightarrow R$ such that $h$ is ...

**3**

votes

**1**answer

95 views

### Intersection of Maximal Left Ideals with Finite Dimensional Quotient

Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I ...

**4**

votes

**0**answers

176 views

### A doubt from the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring”

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff ...

**19**

votes

**1**answer

460 views

### Kaplansky's unit conjecture and unique products

There are three conjectures on group rings that bear the name of Kaplansky (see for example this question). The 'unit conjecture' in the title of the present question is the strongest of them, and ...

**2**

votes

**1**answer

159 views

### Vertices of a polytope as algebra generators

I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...

**3**

votes

**2**answers

324 views

### Finitely generated projective = finitely presented flat over a noncommutative Noetherian ring

Let $R$ be a possibly noncommutative left Noetherian ring and $M$ an $R$-module. I am looking for a reference or a proof for the following fact: $M$ is finitely generated and projective if and only if ...

**13**

votes

**2**answers

736 views

### Are there only finitely many associative algebras of fixed dimension?

Given an algebraically closed field $F$, for any positive integer $n$, are there always only finitely many non-isomorphic (noncommutative) associative algebras (possibly without identity) with ...

**3**

votes

**1**answer

295 views

### How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...

**2**

votes

**0**answers

159 views

### C*-algebras and bounded relations

I'm trying to get used to the language of generators and relations for C*-algebras through Loring's "Lifting Solutions to Perturbing Problems in C*-Algebras". So far this is what I got from the first ...

**5**

votes

**1**answer

204 views

### Injective flat module

Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such.
...

**1**

vote

**0**answers

224 views

### Global dimension of a subalgebra with all units

(All rings here are always assumed to be unital and associative).
Setup
Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying:
If $u$ is a unit ...

**3**

votes

**0**answers

130 views

### Finding an integral basis for the lattice of the form $\mathbb{Z}^J \cap \mathbf{p}^{\perp}$

Let $\mathbf{p}$ be a primitive point in the lattice $\mathbb{Z}^J$
and denote the $J-1$ dimensional vector space $V = \mathbf{p}^{\perp} \subseteq \mathbb{R}^J$.
Let $\Lambda' = \mathbb{Z}^J \cap V ...

**8**

votes

**0**answers

198 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
...

**0**

votes

**0**answers

101 views

### Grading a module

Let $R = \bigoplus_{n\in\mathbb{Z}}R_n$ be a graded ring. I'm trying to understand the structure of simple graded $R$-modules.
In C. Nastasescu and F. Van Oystaeyen book, Methods of graded rings, ...

**1**

vote

**1**answer

129 views

### Modules “projective in a subcategory”

In my research I have come up with the following notion which I would like to learn more about. It may be very naive.
Let $R$ be a ring, $M$ an $R$-module and $S$ a class or $R$-modules closed under ...

**4**

votes

**2**answers

181 views

### Is there a purely module theoretic characterization of semiprimitive rings?

A ring (say unital for simplicity) is semiprimitive (or Jacobson semisimple) if its Jacobson radical is trivial, or equivalently it has faithful semisimple module. Semiprimitivity is a Morita ...

**3**

votes

**1**answer

199 views

### On the socle of rings

Is it possible that the socle of a ring (with identity) is cyclic as a left ideal but not finitely generated as a right ideal !?

**3**

votes

**2**answers

373 views

### Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...

**3**

votes

**1**answer

166 views

### What do epimorphisms in noncommutative rings look like?

The question I want to ask is inspired by this mathoverflow post about epimorphisms in the category of commutative rings. I found the seminar (by P. Samuel) referenced by David Rydh particularly ...