# Tagged Questions

The noncommutative-algebra tag has no wiki summary.

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### 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?

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### $Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$

I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say
...

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### noncommutative polynomials equality

Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$.
Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the ...

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### Units in a group algebra

Let k be a field and let G be a finite group. I would like to know if there is any nice description of the group of units in the group algebra kG. (If there is no nice answer in this generality, ...

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### 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 ...

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### $\mathrm{rk}_R M$ vs $\mathrm{rk}_S M$ - how nice need $R,S$ be?

Let $R\hookrightarrow S$ be Noetherian (noncommutative) rings without zero divisors with $\mathrm{rk}_{R} S < \infty$ (e.g. $S=R*G$ the crossed product of $R$ with a finite group $G$). Let $M$ be a ...

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### For $G=\mathbb{Z}^2\rtimes \mathbb{Z}$, $Spec(\mathbb{Z}G)$=?

Let $G$ be the group $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}=\langle y,z\rangle\rtimes_{\sigma}\langle x\rangle$, where $\sigma(x)=\begin{pmatrix}a, b\\c,d\end{pmatrix}\in SL_2(\mathbb{Z})$, which ...

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### 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 $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...

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### How big can a commutative subalgebra of Weyl algebra be?

Consider the smallest Weyl algebra $A_1=\{q,p; qp-pq=1\}$. It is known that there exist pairs of commuting elements, say $L$ and $M$, that obey various polynomial relations, e.g. elliptic curves. I ...

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### Dimension of commutative subalgebras of a central simple algebra

let $k$ be a field, and let $A$ be a central simple $k$-algebra over $k$.
What is the maximal dimension of a commutative $k$-subalgebra of $A$?
If $A=M_r(D)$, where $D$ is a central division ...

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257 views

### Conjugate linear maps between $*$-algebra modules

Let $A$ be a $*$-algebra, $E,$ and $F$ two $A$-modules, and a map $f:E \to F$ such that
$$
f(ae) = a^*f(e), ~~~~~~~ a \in A.
$$
This seems to me to be the natural generalisation of a conjugate linear ...

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728 views

### Structure theorem for finite dimensional $C^*$-algebras and their representations

I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere.
Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an ...

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### $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...

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### Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
What is the volume of the largest symmetric convex subset ...

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### Normal regular sequence in noncommutative algebras

Does anyone know anything about the normal regular sequences in the quantum plane?
Here are the definitions:
Normal regular sequence: Let $R$ be a ring (not necessarily commutative). A sequence ...

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### Non-simple and non-unital rings with trivial centres

Let $R$ be an associative and non-unital ring. (Suppose that $R$ is $s$-unital, i.e. for each $x\in R$ there is $u,v\in R$ such that $ux=xv=x$.)
It is not difficult to show that if $R$ is a simple ...

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298 views

### Gerstenhaber bracket out of $L_\infty$ algebras

Given a Lie algebra g, with $Ug$ being its universal enveloping algebra, one can construct a cochain complex $d: Ug^n \rightarrow Ug^{n+1}$, and a Gerstenhaber bracket on $\oplus_n Ug^n$ so that ...

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### 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 ...

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### Applications of Govorov-Lazard Theorem?

I asked this question on SE a long time ago, but never received an answer:
The Govorov-Lazard Theorem states that a (left) module over an unital ring is flat iff it is a direct limit of finitely ...

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### 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 ...

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### 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 ...

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### Algorithmically finite-dimensional (noncommutative) algebras.

Can anyone help to find some information about these structures?

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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 ...

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### Properties of ring epimorphisms that are true only over commutative rings

I'm interested in knowing/collecting some properties of epimorphisms of rings (with identity) that are true over commutative rings but are false in the non-commutative case.
Example: I learned from ...

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277 views

### Projectives in the category of discrete G-modules

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5).
Since the cited ...

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### 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 ...

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### 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 $l(I)+l(J)=l(I\cap J)$, ...

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### Example of computation of moduli space of $n$-pointe modules?

I am looking for an example of computation of the isomorphism classes of $n$-point modules over a non-commutative generated graded algebra (assuming all good properties such as Noetherian property). ...

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### Epimorphisms and free submodules

By inspecting the accepted answer to this question
Are epimorphisms from a division ring isomorphisms ?
one obtains the following necessary condition for epimorphisms:
Let $R \le S$ be rings ...

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### Are epimorphisms from a division ring isomorphisms ?

According to Corollary 1.2(3) of the paper Silver: Noncommutative Localizations and Applications. J. of Alg. 7(1964), 44-67:
If $R$ is a (commutative) field and $\alpha: R \to S$ an epimorphism in ...

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### 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 = ...

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### Are annihilation modules in the quantum torus necessarily principal?

I hope that my question yields some standard fact from (noncommutative) ring theory. In discussions with other graduate students, we have outlined some approaches to tackling the question, but ...

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### 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 ...

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### 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 ...

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### Does Castelnuovo-Mumford regularity hold for this $\mathbb{C}$-algebra$?

Let $R$ be a noncommutative finitely generated $\mathbb{C}$-algebra such that its center $S$ is smooth (in commutative sense) and $R$ is finite over $S$. Is there Castelnuovo-Mumford regularity ...

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### Balanced dualizing complex vs rigid dualizing complex?

In noncommutative projective geometry, there is a counterpart of dualizing complex in commutative world. It seems to me that they are called either a balanced dualizing complex or rigid dualizing ...

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### what is the definition of the Picard group of a (non necessarilly commutative) Ring?

Hi. I have only able to find the definition of $Pic(R)$ for a commutative ring $R.$ Which is the isomorphism classes of projective $R$-modules of rank $1,$ and the product given by $[A][B]=[A\otimes_R ...

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### Point modules of quantum projective space $\mathbb{P}^n$

Let $A$ be a quantum $\mathbb{P}^n$ defined by
$$
A=\mathbb{C}\langle x_1,x_2,\dots,x_{n+1}\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le n+1}.
$$
I would like to know the set $X$ of isomorphism ...

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### Strongly Noetherian property. When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian rings $A$ and $B$?

Let $k$ be a field. It is well-known that $A\otimes_{k}B$ is not necessarily Noetherian even if $k$-algebras $A$ and $B$ are Noetherian. For example $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}$.
When ...

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### 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 ...

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### Moduli space of modules over non-commutative rings

Let $X=Proj(A)$ be a projective scheme, one can the moduli space of coherent sheaves on $X$ with fixed Hilbert polynomial and stability. Since coherent sheaves on $X$ are all obtained as the ...

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### 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 $Ext^{2}(M,M)$. ...

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### Finite dimensional consistently graded Lie superalgebras of depth greater than 2

Victor Kac, in the paper
"Classification of infinite-dimensional simple linearly compact Lie superalgebras", http://www.mat.univie.ac.at/~esiprpr/esi605.pdf
writes at the beginning of section 5 ...

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### Isomorphisms of quantum planes

Let $k$ be a field and $q\in k^{*}$. The quantum plane $k_{q}[x,y]$ is the algebra $k\langle x,y\rangle/\langle xy=qyx \rangle$ (i.e. the quotient of the free non-commutative $k$-algebra on two ...

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### Liftability of a submodule from an associated graded module

Let $k$ be a field, $A$ a $k$-algebra (probably noncommutative), and $M$ an $A$-module that's finite-dimensional as a vector space over $k$.
Let $Gr(M;k)$ denote the set of all $k$-subspaces of $M$, ...

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### 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 ...

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### Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. I'm seeking a kind of Schur's lemma, with $\mathrm{Hom}_R (M,N)$ replaced by $M \otimes_R N$. So my questions are:
Can ...

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### 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 ...

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### Classification of long exact sequences

Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.
The category $\mathcal C$ is naturally additive as a subcategory of ...

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### Projective dimension over hypersurface

Let $R$ be (not necessarily commutative) ring and $S$ a simple right $R$-module. Let $f\in Ann(S)$ be normalizng and a non-zero divisor. Is it always true that
$$
pdim_{R}(S)=pdim_{R/(f)}(S)+1?
$$