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4
votes
1answer
752 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 ...
23
votes
1answer
687 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 ...
6
votes
1answer
277 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
vote
1answer
247 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
1answer
277 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 ...
8
votes
1answer
510 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 ...
2
votes
2answers
240 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 ...
4
votes
0answers
149 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
2answers
709 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 ...
10
votes
2answers
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 but I'm ...
9
votes
2answers
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 ...
46
votes
2answers
4k 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
3answers
928 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
2answers
849 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
vote
1answer
202 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 ...
7
votes
1answer
334 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 ...
1
vote
0answers
585 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 ...
3
votes
1answer
220 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 ...
9
votes
1answer
562 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 ...
12
votes
4answers
2k 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, ...
11
votes
2answers
1k 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 ...
13
votes
2answers
791 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 ...
8
votes
3answers
895 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 ...
11
votes
5answers
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 ...
2
votes
1answer
219 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). ...
3
votes
4answers
706 views

Homological dimension of a graded ring which is like polynomial ring

Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation ...
7
votes
1answer
290 views

Linear disjointness of subfields of a centrally finite division algbera

I am looking for papers or books which discuss this problem. Thank you for reading: Let K and L be two subfields of a non-commutative division algebra D with the center Z. Suppose that K and L ...
12
votes
1answer
455 views

Tensor products and two-sided faithful flatness

Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both sides, so $M \to M ...
5
votes
2answers
659 views

Is there a name for this algebraic structure?

I found myself "naturally" dealing with an object of this form: X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and ...
12
votes
2answers
1k views

What properties “should” spectrum of noncommutative ring have?

There are already a lot of discussion about the motivation for prime spectrum of commutative ring. In my perspective(highly non original), there are following reasons for the importance of prime ...
8
votes
3answers
578 views

Existence of non-commutative desingularizations

Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as a weak ...
10
votes
0answers
666 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 ...
6
votes
2answers
333 views

Eigenvalues of an element in a Weyl algebra

I have an operator acting on the polynomial algebra $\mathbb{C}[x,y,z]$ that I would like to find the eigenvalues/eigenvectors of. More specifically, let $P(x_1, \ldots, x_6)$ be a homogeneous ...
4
votes
1answer
293 views

Non-commutative versions of X/G

Let $X$ be a Riemannian manifold and let $G$ be a (at most countable, if that matters) discrete group acting properly and by isometries on $X$. Let $\mathcal{O}$ be the sheaf of analytic functions on ...
4
votes
1answer
372 views

Depth Zero Ideals in the Homogenized Weyl Algebra

Let $\mathcal{D}$ be the $n$th Weyl algebra $ \mathcal{D} :=k[x_1,...,x_n,\partial_1,...,\partial_n] $, where $\partial_ix_i-x_i\partial_i=1$. Let $\widetilde{\mathcal{D}}$ be its Rees algebra, ...
5
votes
1answer
242 views

Classifying Algebra Extensions over a fixed extension?

There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that ...
4
votes
1answer
146 views

When is the essential extension commutes with colimits(or push forward)

Let $M$ be an $R$-module,where $R$ is a hereditary (or cohomological dimension less or equal to 1).Take $E(R)$ to be injective hull of $R$, then we have the essential extension $i:R^I\rightarrowtail ...
13
votes
1answer
772 views

Graded commutativity of cup in Hochschild cohomology

I am trying to get used to Hochschild cohomology of algebras by proving its properties. I am currently trying to show that the cup product is graded-commutative (because I heard this somewhere); ...
2
votes
1answer
112 views

Some equivalent statements about primitive algebras

I was reading a paper, and it said that the following were equivalent using the Axiom of Choice, but I tried working it out, and I wasn't sure how: an algebra $A$ is primitive; $A$ has a proper left ...
13
votes
9answers
1k views

Examples of noncommutative analogs outside operator algebras?

Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include: A $C^\ast$-algebra is a ...
8
votes
4answers
642 views

Morita equivalence and moduli problems

Two rings $A$ and $B$ are said to be Morita equivalent if the category of modules over $A$ and $B$ are equivalent as additive categories. (Here I'm considering left modules). Ex: $M_n(R)$ (the algebra ...
6
votes
3answers
235 views

Inverses in convolution algebras

Let $G$ be a locally compact totally disconnected group, and to make life easy let's suppose its Haar measure is bi-invariant. Let $C_c(G)$ be the space of locally constant complex functions on $G$ ...
12
votes
1answer
663 views

Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complex-valued functions on its ...
1
vote
3answers
1k views

Various Cartan's Lemmata

I am a bit amazed by "Cartan's Lemma".. I have so far seen it in : Algebraic Geometry sources: Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...
7
votes
5answers
1k views

Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring

Can anyone prove that a Weyl Algebra is not isomorphic to a matrix ring over a division ring?
2
votes
2answers
870 views

An “Elementary” Math Question Generalized (Ring Theory Perhaps)

The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics" "Prove that if integers a_1, ..., a_n are all distinct, then the ...
7
votes
2answers
566 views

Do torsion-free groups give projectionless group ($C^\ast$) algebras?

One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...
21
votes
3answers
1k views

When does the converse to Schur's Lemma hold?

Let $R$ be a commutative ring, let $A$ be an $R$-algebra, and let $M$ be an $A$-module. If $M$ is simple, then End$_{A-mod}(M)$ is a division ring. A common use is when $R$ is the complex numbers ...
12
votes
2answers
920 views

How much theory works out for “almost commutative” rings?

I've been reading about D-modules, and have seen a proof that D_X, the ring of differential operators on a variety, is "almost commutative", that is, that its associated graded ring is commutative. ...
7
votes
1answer
387 views

Maximal localizations of von Neumann algebras

Suppose M is a von Neumann algebra. Denote by L its maximal noncommutative localization, i.e., the Ore localization with respect to the set of all left and right regular elements, i.e., elements whose ...