The noncommutative-algebra tag has no usage guidance.

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### An example where finitistic dimension does not equal right global dimension?

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...

**6**

votes

**2**answers

582 views

### Simple Ore extensions

Let $R[x;\sigma,\delta]$ be an Ore extension, where $R$ is an associative and unital ring and $\sigma : R\to R$ is a (not necessarily injective!) ring endomorphism. (In the literature it is often ...

**6**

votes

**1**answer

276 views

### A potential resolution of $R/r$

The DGA
For $k$ some field, let $R$ be a $k$-algebra, and let $r\in R$.
Define a differential graded algebra $\mathbf{R}_r$ as follows. As a graded algebra, it is isomorphic to $R\langle t\rangle$, ...

**2**

votes

**0**answers

289 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 ...

**20**

votes

**8**answers

4k views

### Simplest examples of rings that are not isomorphic to their opposites

What are the simplest examples of
rings that are not isomorphic to their
opposite rings? Is there a science to constructing them?
The only simple example known to me:
In Jacobson's Basic Algebra ...

**2**

votes

**2**answers

404 views

### Model Theoretic Localization

This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask.
1) Let $\sigma = (A; \{0,1\}; +, ...

**9**

votes

**1**answer

2k views

### Are there any finitely generated artinian modules that are not Noetherian?

It is well known that for rings, Artinian implies Noetherian (the famous Hopkins–Levitzki theorem) and it is also well known that there are Artinian modules which are not Noetherian. A simple example ...

**2**

votes

**2**answers

540 views

### Torsion in tensor products over noncommutative rings

I know that the problem of torsion in tensor products, even of torsion free modules, is a very delicate thing. Unfortunately i don't have a deeper insight into this subject, so i don't know how things ...

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vote

**0**answers

394 views

### The sum of a nilpotent left ideal and a nil left ideal

In class, we recently saw that the sum of 2 two-sided nil ideals is a nil ideal. We were asked to show that the sum of a niplotent left ideal and a nil left ideal is a nil left ideal.
I am having ...

**1**

vote

**1**answer

235 views

### Ore Extensions and the Construction of the Quantum General Linear Group

In the usual (fomal) construction of the quantum general linear group $GL_q(N)$, an Ore extension is used. See for example Kassel. Why is this necessary? Surely one can just augment the set of ...

**4**

votes

**2**answers

247 views

### unitary reduction of $q$-normal matrices

The unitary reduction of normal matrices is a well-known fact: if $A\in M_n(\mathbb C)$ commutes with its Hermitian adjoint $A^*$, then there exists a unitary $U\in\mathbb U_n$ and a diagonal matrix ...

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votes

**8**answers

3k views

### What makes a theorem *a* “nullstellensatz.”

I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...

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votes

**2**answers

2k views

### Jacobson radical = intersection of all maximal two-sided ideals

I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can ...

**4**

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**0**answers

620 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 ...

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votes

**2**answers

392 views

### on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring

Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some ...

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**2**answers

608 views

### Unit ideal in non-commutative rings [closed]

In a non-commutative ring (with identity), is it possible for an element which does not possess left or right inverses to generate the entire ring? i.e. $(r)=R$, where (r) is the two-sided ideal ...

**3**

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**1**answer

322 views

### Finite Homological Dimension of R/P for all P for module finite non-commutative rings

I have a reasonably precise question which I hope is clear enough to get a nice answer. Let R be a Noetherian non-commutative ring which is finite as a module (and flat/free if it helps) over it's ...

**11**

votes

**2**answers

1k views

### Why does the Grothendieck group $K_0(R)$ of a ring not depend on our choice of using left modules instead of right modules?

I am under the impression that in the definition of the Grothendieck group $K_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K_0$ construction to the exact category of ...

**10**

votes

**2**answers

2k views

### Compute Lie algebra cohomology

Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation? What if the Lie algebra is finite dimensional?
In my case I would like to be able to ...

**0**

votes

**3**answers

1k views

### How to work with co-multiplication?

Let $C$ be a coalgebra and $\Delta: C \to C\otimes C$ a co-multiplication map. Then, due the co-associative property we can consider $\Delta^m$. But how is defined $\Delta^{m}: C \to C^{\otimes m}$???
...

**-1**

votes

**2**answers

256 views

### Identifying a sequence of polynomials

Studying a specific quantum cluster algebra, I have come across the following sequence of polynomials :
$$X_1$$
$$q^{-1/2}(X_1X_2-1)$$
$$q^{-1/2}(X_1X_2X_3-X_3-X_1)$$
...

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votes

**1**answer

263 views

### Does there exist any massive proper $C^*$-subalgebra?

Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff
1. $A$ is a subalgebra of $B$;
2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...

**3**

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**0**answers

155 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 $sr'=r$, ...

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votes

**3**answers

690 views

### Set of invertible operators in B(H) is connected. Is it true? Is there a reference?

Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...

**1**

vote

**1**answer

274 views

### Morphisms of a simple sheaf over an algebra to its double dual

Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O_S$-module of finite rank. Let $M$ be a coherent and ...

**4**

votes

**1**answer

866 views

### Length of a module over different rings

Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.
Under what ...

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**1**answer

742 views

### Idempotents in Rings of Differential Operators

Differential Operators on General Commutative Rings
Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator ...

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**1**answer

299 views

### Simultaneously orthogonally transform two SPD matrices to tridiagonal form?

Supposing you have two SPD matrices $A,B\in\mathbb{R}^{n\times n}$ are there any known results on the existence or non-existence of a unitary matrix $Q$ such that $Q^\top A Q=T_A$ and $Q^\top B Q=T_B$ ...

**1**

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**1**answer

255 views

### Q-Divisor and Determinant Map on a Maximal Order

Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring.
Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the ...

**0**

votes

**1**answer

283 views

### Uniqueness of maximal subfields

Let D be a division ring with center Z. Let R and K be two maximal subfields of D, both purely inseparable of exponent one ( means the p power of each of them in Z).
Why are R and K isomorphic?
Or a ...

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**1**answer

550 views

### Mathematical software for computing in integral group rings of discrete groups?

I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where ...

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votes

**2**answers

256 views

### How canonical is the triangular decomposition of a rational Cherednik algebra?

Introduction:
Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, let $G \leq \mathrm{GL}(V)$ be a finite subgroup and let $\kappa:V \times V \rightarrow \mathbb{C}G$ be an alternating bilinear ...

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152 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 ...

**4**

votes

**2**answers

780 views

### Semiprime (but not prime) ring whose center is a domain

The center of a prime ring is a domain and the center of a semiprime ring is reduced.
Now I have no evidence to believe that if the center of a semiprime ring R is a domain,
then R has to be a ...

**12**

votes

**2**answers

1k views

### Maximal ideal that annihilates entire ring

Does there exist a ring $R$ with a nonzero maximal ideal $M$ such that $R^2=R$ and $MR = RM = 0$?
Here $R$ is associative but does not have an identity (obviously). It seems a simple enough question ...

**12**

votes

**2**answers

2k views

### Left and right eigenvalues

A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...

**51**

votes

**2**answers

5k views

### How would you solve this tantalizing Halmos problem?

1-ab invertible => 1-ba invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric series. In ...

**16**

votes

**3**answers

997 views

### What is the precise relationship between groupoid language and noncommutative algebra language?

I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose:
objects are groupoids;
1-morphisms are (left-principal?) bibundles;
2-morphisms are bibundle ...

**3**

votes

**2**answers

912 views

### Spectral decomposition for an arbitrary linear combination of position and momentum operators

Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by:
Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn)
Pi ψ(q1,q2,...,qn) = -i ...

**1**

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**1**answer

213 views

### Algebra of endomorphisms of f.g. modules as subquotients of matrix algebras

Let $A$ be a $C$-algebra, where $C$ is a commutative ring with $1$, and $M$ be a finitely generated left $A$-module.
Question: Is it true that we can always find a positive integer $n$, a ...

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**1**answer

342 views

### Semisimple-ish rings!

Let S be the class of all rings R which have 1 and satisfy this condition:
for every "non-zero" right ideal I of R there exists a "proper" right ideal J of R such that I + J = R. (The + here is not ...

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**0**answers

605 views

### What is a “double star-product”

Michel Van den Bergh introduced the notion of a double Poisson algebra. The definition is cooked up such that the representation varieties of such an algebra are Poisson varieties.
Is there a notion ...

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**1**answer

225 views

### Special subalgebras of central simple algebras

In this question F is a field and all algebras are finite dimensional F algebras.
Let X be the set of all F algebras A for which there exist an F algebra B and an F division algebra D such that F is ...

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**1**answer

587 views

### Division algebras in which every proper subfield is maximal

I have a (noncommutative) division algebra D which is finite dimensional over its center F. I know that every subfield of D which contains F properly is a maximal subfield of D. What can we say about ...

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3k views

### What are your favorite finite non-commutative rings?

When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.
There are many convenient examples of commutative rings, both finite and infinite, ...

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2k views

### when are algebras quiver algebras ?

Good Morning from Belgium,
I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a ...

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**2**answers

839 views

### A Non-Commutative Nullstellensatz

In studying presentations of pro-$p$-groups via generators and relations, one is led (via the so-called Magnus embedding) to questions involving power series in non-commuting variables. Results from ...

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**3**answers

957 views

### construct scheme from quivers?

I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...

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**6**answers

2k views

### Definition of an algebra over a noncommutative ring

I've tried in vain to find a definition of an algebra over a noncommutative ring. Does this algebraic structure not exist? In particular, does the following definition from ...

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**1**answer

226 views

### Formal deformations of algebras over not necessarily commutative rings

In Iain Gordon's survery article "Symplectic reflection algebras" the concept of formal deformations of algebras over semisimple artinian (not necessarily commutative) rings is summarized (chapter 2). ...