Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups.

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21
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2k views

Sums of injective modules, products of projective modules?

Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension? Analogously, under what assumptions on R does ...
16
votes
0answers
246 views

Kaplansky's unit conjecture and unique products

There are three conjectures on group rings that bear the name of Kaplansky (see for example this question). The 'unit conjecture' in the title of the present question is the strongest of them, and ...
14
votes
0answers
251 views

Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google: Suppose I have a countable field, $k$. ...
12
votes
0answers
523 views

How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
12
votes
0answers
570 views

Countable Hom/Ext implies finitely generated

Today I learned this interesting fact from Jerry Kaminker: If $A$ is an abelian group such that $\mathrm{Hom}(A,\mathbb{Z})$ and $\mathrm{Ext}(A,\mathbb{Z})$ are both countably generated, then in fact ...
11
votes
0answers
501 views

Given an algebra, can it be realized as a block of a Hopf algebra?

During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in ...
9
votes
0answers
108 views

Detecting invertible elements in group rings by their images for finite quotients of the group

Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent). Consider an element ...
9
votes
0answers
258 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 ...
9
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483 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 ...
9
votes
0answers
227 views

What do Multilinear Forms tell us about Representations?

The last few days I have been calculating whether certain group representations are real, complex, or quaternionic. It is well-known that the type of the representation corresponds to what type of ...
8
votes
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147 views

Depth for non-commutative rings

The depth of a ring or module is one of the most basic invariants in commutative ring theory. Q1: Is there also a powerful notion of depth for non-commutative rings ? By a search in mathscinet, I ...
8
votes
0answers
460 views

Skew polynomial algebra

When I was a very little hare, a big grey wolf told me about the following skew polynomial algebra, which I never understood. My question is whether the following construction is a part of some bigger ...
7
votes
0answers
193 views

What happens to simple modules under Ringel duality?

If $A$ is a quasi-hereditary algebra then its Ringel dual $A'=End_A(T)$ is the endomorphism algebra of a (minimal) full tilting module $T$ for $A$. The algebra $A'$ is again quasi-hereditary, although ...
7
votes
0answers
163 views

Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?
7
votes
0answers
129 views

Is 2 a zerodivisor in the ring parametrizing rank-n algebras?

Let $n$ be a natural number; I am studying (commutative) rings $R$ and $R$-algebras $A$ such that $A\cong R^n$ as $R$-modules. There is a universal such algebra: a ring $R_0$ and a free, rank-$n$ ...
7
votes
0answers
327 views

Is the class of additive groups of rings axiomatizable?

I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...
7
votes
0answers
483 views

duality between universal enveloping and function algebra for GL(n)

Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular ...
6
votes
0answers
146 views

Series in topological rings that only converge if almost all summands are zero

While trying to understand a certain topological ring better, I stumbled onto the following question. Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...
6
votes
0answers
148 views

Is it possible to define Hopf algebra as an object in the category of (associative) algebras?

I think this must be well-known, but I can't find references, so my apologies from the very beginning. Consider first the notion of bialgebra. It is usually defined as an object $B$ in the category ...
6
votes
0answers
238 views

Descending chain condition in noncommutative rings

By Hopkins Theorem it is well-known that every right (resp. left) artinian unitary ring is right (left) noetherian. Suppose that a noncommutative unitary ring R satisfies the descending chain ...
6
votes
0answers
132 views

Classifying algebras with two idempotent generators and involution

Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$. For ...
6
votes
0answers
352 views

Ideal class group isomorphic to $\mathbb{Z}$

Hi everybody, I wonder if someone could provide me with a simple example of a Dedekind ring whose ideal class group is isomorphic to $\mathbb{Z}$. The point is I would like the example simple enough ...
6
votes
0answers
377 views

Can we write unitary matrices as positive linear combinations of Hermitian matrices?

The space $M_n:=M_n(\mathbb{C})$ of complex $n\times n$ matrices has the structure of a finite-dimensional complex vector space. The space of Hermitian matrices forms a cone in this vector space ...
6
votes
0answers
326 views

Quantum polynomial rings and singularities

Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...
5
votes
0answers
203 views

A generalization of real characters on a group

Yesterday I understood that I can't live without this construction: Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps ...
5
votes
0answers
151 views

Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients. $$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$ Monomials $x^k$ are mapped to $n ...
5
votes
0answers
122 views

Software for BMW algebra calculations?

Does software exist for computations in the BMW algebra? For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...
5
votes
0answers
322 views

The logarithm over $\mathbb F_1$

In 'Cyclotomy and analytic geometry over F1', Manin proposes a version of the notion of `analytic function' over the 'field with one element $\mathbb F_1$'. Question 1: can somebody explain or give ...
5
votes
0answers
131 views

Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that: $\mathcal C$ is ...
5
votes
0answers
200 views

On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix. Suppose there exists ...
5
votes
0answers
197 views

How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two. Version 1: Let $(C,\otimes)$ be any monoidal ...
5
votes
0answers
253 views

Homological dimension of completed path algebras.

Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations. Is it true that the I-adic completion of A has finite homological dimension?
5
votes
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211 views

Cocontinuous monadic functors

Some forgetful functors of algebraic categories preserve colimits, but most do not. In order to understand this phenomen in general, I have classified cocontinuous monadic functors whose target is an ...
5
votes
0answers
319 views

Generators of associated graded algebra

Suppose that $A = \bigcup_{n=0}^{\infty} A_n$ is a filtered algebra over a field $k$. The associated graded algebra is $\mathrm{gr} A = \bigoplus_{n=0}^{\infty} A_n/A_{n-1}$, where we define $A_{-1} ...
5
votes
0answers
404 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
99 views

Does Mittag-Lefflerness descend?

I have read in the Stacks Project that if $A \to B$ is a faithfully flat ring homomorphism, $M$ is an $A$-module, and $M \otimes_A B$ is a flat, Mittag-Leffler $B$-module, then $M$ is a flat, ...
5
votes
0answers
143 views

integral versus adjoint action on Hopf algebra

Suppose that $H$ is a finite dimensional Hopf algebra (with counit $\varepsilon$) and $T$ is a non zero right integral of $H^{\star}$ (the dual Hopf algebra). Let $ad_h$ be the adjoint action on $H$, ...
5
votes
0answers
243 views

Testing isomorphism of finitely generated algebras

Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and $C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated ...
5
votes
0answers
220 views

Roots of unity in algebraic K-theory

For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit. For ...
5
votes
0answers
86 views

Ring properties which can be checked on sufficiently many Ore localizations

Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of ...
5
votes
0answers
321 views

Lie locally nilpotent associative algebras

Let $A$ be an associative algebra over a field. Then $A$ can be regarded as a Lie algebra via the Lie bracket defined by $[a,b]=ab-ba$ for every $a,b\in A$. The algebra $A$ is called Lie locally ...
5
votes
0answers
275 views

Is there an idempotent measure on the free LD system?

This is a follow up question to MO question "Idempotent measures on the free binary system?". Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law: ...
5
votes
0answers
386 views

localization at central maximal ideals

If M is a maximal ideal of Z(R), the center of a ring with identity R, and R_M, the localization of R at M, is a commutative field, what can we say about R? My guess is that we can's say that much ...
5
votes
0answers
304 views

is there a notion of weakly noetherian?

A left module M over a ring is finitely presented iff the functor Hom(M,_) preserves filtered inductive limit. Meanwhile, if M is finitely generated, then for every filtered system { N_i } of left ...
5
votes
0answers
436 views

Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups

In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...
4
votes
0answers
153 views

A doubt from the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring”

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff ...
4
votes
0answers
73 views

Does this kind of non-noetherian bimodule exist?

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 $S$-module, ...
4
votes
0answers
118 views

Formal DG-algebras

Sorry for this question but I really have difficulties with model categories. Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...
4
votes
0answers
95 views

Sum of projective submodules of a projective over a semihereditary ring

Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a ...
4
votes
0answers
226 views

Ext groups of affine scheme

Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and $\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$ respectively. I ...