The noncommutative-algebra tag has no wiki summary.

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### Motivation for the Preprojective Algebra

Let $Q=(Q_0,Q_1)$ be a quiver and $k$ a field. We construct a new quiver $\bar{Q}$ in the following way. Let the vertices of $\bar{Q}$ be the same as the vertices of $Q$, and let the arrows of ...

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### 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?

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### 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 PoincarĂ© duality?
All i need to find is a protective ...

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### For Finite Dual when is $(A \otimes A)^o = A^o \otimes A^0$?

Let $A$ be any $k$-algebra. The finite dual or restricted dual of $A$ is
$$
A^o = \{f \in A^* ~ | ~ f(I)= 0, \text{ for some ideal } I \subseteq A, \text{ such that } \text{dim}_k(A/I) < \infty\}.
...

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### Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$.
Explicitly how would one go about computing ...

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### Jacobi-Zariski exact sequence question

Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ ...

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### Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...

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### Hochschild homology of quiver algebras

Let $K$ be a field and $\Gamma$ a quiver (=multidigraph) and $K[\Gamma]$ its quiver algebra (free $K$-module on the set of all paths of length $\geq0$ where multiplication is concatenation if ...

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### Explicit isomorphism for quaternion algebras over $\mathbb{Q}$?

It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes_K K_v$ is a division ...

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### Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$?
Is there any ...

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### Integral elements of quaternion algebras with predescribed properties

In the course of doing some calculations I have found myself wanting to answer the following question:
Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...

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### Why is “naive” definition of non-commutative spectrum bad?

It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative ...

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### 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 ...

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### Computing noncommutative geometries

I find myself needing to construct some noncommutative geometries. I want to take various (algeba-) geometric objects and look at their noncommutative analogs. Is there a constructive way to do this? ...

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### Problem with Smoothness and quasi-freeness

Let A be a unital associative algebra over a field k.
Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...

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### Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left)-noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by a finite set $x_1,\dots,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. A Poisson ...

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### 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 ...

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### When is the dual module isomorphic to conjugate module of a *-algebra

Let $A$ be a $\ast$-algebra, and let ${\cal M}$ be a module over $A$. As usual, we denote the dual module of $M$ by $M^*$. Moreover, we denote the conjugate module of $M$ by $\overline{M}$, where as ...

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### Variety of factorizations of differential operator

Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let ...

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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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### 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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### 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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### 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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### 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 = ...