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17
votes
0answers
537 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
12
votes
0answers
566 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 ...
11
votes
0answers
369 views

Nullstellensatz for quaternionic plane curves?

By a quaternionic plane curve I mean the zero locus of a noncommutative polynomial in two variables, $x$ and $y$ say, over ${\Bbb H}$, Hamilton's quaternions. It is evidently well-known that, after ...
10
votes
0answers
1k views

Is my definition of a context algebra new?

In my DPhil thesis, I defined what I called a context algebra as a model of meaning in natural language. The idea is to mathematically formalise the notion that meaning is determined by context. It ...
9
votes
0answers
325 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 ...
8
votes
0answers
136 views

Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying $a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows. ...
8
votes
0answers
209 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 $...
6
votes
0answers
79 views

Center Picard group non-commutative algebra

I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra. Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by $\mathrm{Pic}(A)$...
6
votes
0answers
161 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 ...
5
votes
0answers
90 views

Differentially closed fields

Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$. Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...
5
votes
0answers
382 views

Central Element in Sklyanin Algebras?

I'm interested in Sklyanin Algebras or Artin-Shelter regular algebras of type A. These are generated in degree 1 by three variables x,y,z, and have three defining relations in degree 2, which you can ...
5
votes
0answers
418 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
0answers
640 views

Commutator Baker-Campbell-Hausdorff formula

Consider the Baker-Campbell-Hausdorff formula $\Phi(X,Y)\in\mathbb{Q}\langle\!\langle X,Y\rangle\!\rangle$ in non-commutative variables. Define $X*Y:=\Phi(X,Y)$ and $[X,Y]=(-X)*(-Y)*X*Y$, and then (as ...
4
votes
0answers
63 views

An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. Is ...
4
votes
0answers
189 views

Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
4
votes
0answers
99 views

Nilpotent operator of the Weyl algebra

For a research project I'm currently working on, I came across the following problem: Let $A=$ $^{k <x,y> }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $p$, where $...
4
votes
0answers
185 views

Discriminants of Clifford algebras

I have a Clifford algebra defined over a field of characteristic not equal to $2$. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...
4
votes
0answers
152 views

Injectivity criterion for surjective coalgebra maps: does it hold in full generality?

Let $k$ be a commutative ring. Let $C$ be a filtered $k$-coalgebra. This means a $k$-coalgebra equipped with an increasing $k$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...
4
votes
0answers
639 views

Generalized Courant-Fischer theorem

Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $...
4
votes
0answers
156 views

Global dimensions of orders over non-Gorenstein centers

This question concerns the following Lemma 4.2 in this paper by Van den Bergh: Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\...
3
votes
0answers
109 views

When a ring is a polynomial ring?

In the paper (2.11) the authors show that if $k^*$ is a separable algebraic extension of $k$ and $x_1,x_2, \ldots, x_n$ are indeterminates over $k^*$ and a normal one dimensional ring $A$ with $k \...
3
votes
0answers
77 views

Projective dimension of ring over its center

If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite? (Assuming that $A\neq Z(A)$).
3
votes
0answers
191 views

Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let $M_1,...
3
votes
0answers
166 views

AS Cohen Macaulay algebras and dualizing complexes

Let $A$ be an $\mathbb N$-graded algebra such that $A_0 = k$ is a field. This are usually called graded connected algebras. One can define a torsion functor with respect to the ideal $\mathfrak m = \...
3
votes
0answers
231 views

Deformation of modules over noncommutaitve rings

Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(M,M)$ and the obstruction is parametrized by $Ext^{2}(M,M)$. ...
3
votes
0answers
455 views

Noncommutative analog of Koszul complex

Let $R=R_0 \oplus R_1 \oplus R_2 ...$ be a graded not necessarily commutative algebra over field $k$ and $R$ is generated by $R_1$, $R_0=k$. In commutative situation if one wants free resolution of $R/...
3
votes
0answers
353 views

Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task: I have two graded (generally ...
3
votes
0answers
156 views

Pulling out factors in a Noetherian Domain

Let $R$ be a Noetherian domain (not-necessarily commutative), and let $S$ be a Noetherian subring of $R$. An element $r\in R$ is left $S$-irreducible if, for any $s\in S$ and $r' \in R$ with $sr'=r$, ...
2
votes
0answers
66 views

GK dimension of generalized Weyl algebras

I believe that the GK dimension for a generalized Weyl algebra $D(\sigma,a)$ is just the GKdim$(D) + 1$. Does anyone have a reference for this? I can find partial results, and I am sure this is ...
2
votes
0answers
169 views

Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...
2
votes
0answers
82 views

Centers of Noetherian Algebras and K-theory

I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...
2
votes
0answers
114 views

Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$. Q1) Is it ...
2
votes
0answers
100 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
0answers
180 views

Tensor product of commutators vs. commutator in a tensor product

Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and ...
2
votes
0answers
74 views

Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings

In one paper from 1980 I found a note that there are no known algorithms for solving homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring. (The non-...
2
votes
0answers
77 views

Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
2
votes
0answers
58 views

The norm of a polynomial f in a skew polynomial ring must be in the center

This is proved in Prop 1.7.1 in Jacobson's book ``Finite dimensional division algebras over fields". But I am not clear why the norm n(f), defined as the norm of the matrix representation of f by ...
2
votes
0answers
81 views

Fat modules on some algebras.

Let $A$ be a graded $k$-algebra and $M$ a graded right $A$-module. $M$ is called a fat $A$-module if it is generated by degree $0$ and has constant Hilbert polynomial $2$. I wonder for which finitely ...
2
votes
0answers
57 views

Boundedness of modules on AS regular algebras

Let $k$ be an algebraically closed field and $A$ be an Artin-Shelter regular $k$-algebra. Fix a numerical polynomial $H(t)$. I would like to know whether or not semi-stable f.g. graded $A$-modules ...
2
votes
0answers
121 views

Noncommutative Castelnuovo-Mumford regularity

I am looking for noncommutative version of Castelnuovo-Mumford regularity. To be more precise, let $A=\oplus_{i=0}^{\infty}A_{i}$ be a $good$ (finite global dimension, connected etc) noncommutative ...
2
votes
0answers
291 views

Tensor power- Notation question

Hi everyone I have a notational question, which is written usually in papers, but I can not figure it out what could be. Let $M$ be an $A$-module. I have seen this notation $$M^{\otimes -n}$$ I ...
1
vote
0answers
75 views

Is $T(V) \rtimes T(V^* \otimes V)$ a bialgebra?

Let $V$ be a vector space and $V^*$ the dual vector space. Let $T(V)$ be the tensor algebra of $V$. The algebras $T(V)$ and $T(V^* \otimes V)$ are bialgebras. I am trying to find some bialgebra ...
1
vote
0answers
61 views

Completion of an algebra

Based on arXiv:math/9802041v1, there is a definition for $NC$-filtration and $NC$-completion of an associated algebra over the complex numbers: Let $R$ be an associative algebra and $R^{\rm Lie} = (R,...
1
vote
0answers
117 views

Are these quaternion algebras definite or indefinite?

By investigating a different problem I have ended up looking at Quaternion algebras and have a lot to learn about them. Before I do, however, I want to see if my idea has any hope of being useful. So ...
1
vote
0answers
61 views

Turning left modules into right modules over a homotopy Gerstenhaber algebra

For simplicity's sake, let $A$ be a dg-algebra over $\mathbb{Z}/2\mathbb{Z}$. In the case when $A$ is a commutative algebra, we can turn a left $A$ module into a right $A$ module trivially. Of course ...
1
vote
0answers
107 views

Criterion for global dimension of subring

All rings are assumed to be associative and unital. If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...
1
vote
0answers
225 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 ...
1
vote
0answers
144 views

Chinese Remainder Theorem

On a non-commutative ring we have this generalization of CRT. I wonder if there is a statement involving only left or right ideals. do you know any?
1
vote
0answers
166 views

What are the enforceable models of local artinian rings?

I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem ...
1
vote
0answers
214 views

Annihilator ideals

For an ideal $I$ of a ring $R$ with identity, let $r(I)=\{r\in R: Ir=0\}$ and $l(I)=\{r\in R: rI=0\}$. Question: If for any two ideals (two-sided ideal) $I, J$ of $R$, we have $l(I)+l(J)=l(I\cap J)$, ...