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

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7
votes
2answers
231 views

Integration on Compact Semirings

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...
2
votes
0answers
109 views

Noncommutative HKR theorem

What is the analog of HKR theorem in the noncommutative world? Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a smooth commutative algebra $A$ of finite type ...
4
votes
1answer
114 views

Isomorphism of matrix ring over ore domain

Let $R_1,R_2$ be (left and right) ore domains. Does $ Mat_n(R_1)\cong Mat_m(R_2)$ implie m=n and $q.f.(R_1)\cong q.f.(R_2)$? An counter example, a proof or a reference is welcomed. Thanks
4
votes
1answer
228 views

Re-interpreting vector spaces in a choice-less model of ZF as modules over a regular ring in ZFC

I am searching a module $M$ over a (von Neumann) regular ring $A$ ($\forall a\in A$, $\exists x\in A$: $axa=a$) with two properties: (1) every finitely generated submodule of $M$ is projective ...
3
votes
1answer
251 views

Metrics and completions on the direct limit of matrices of all sizes over arbitrary fields

Let $M_n$ denote the $n$ by $n$ matrices. Consider the homomorphisms $$\phi_{n,kn}: M_n \rightarrow M_{kn}$$ which takes a matrix $A \in M_n$ to $A \otimes I_k \in M_{kn}.$ This gives a sensible way ...
1
vote
1answer
110 views

General criterion to find a Z-basis in a fixed generating subset

Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$ be a fixed finite subset. Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the ...
-3
votes
0answers
26 views

On a bijection with a division ring [closed]

Let $K$ be an infinite division ring and $n$ a natural integer . I need to show that there exists a bijection between $K$ and $K^n$. Well, I'm not sure of this, could you help me with it ? A ...
9
votes
1answer
191 views

Motivation behind the definition of hochschild cohomology

For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...
9
votes
2answers
249 views

Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$? To prevent things from being too easy, I ...
4
votes
3answers
685 views

Splitting of a division algebra with an involution of second kind

Let $k$ be a field, $K/k$ a separable quadratic extension, and $D/K$ a central division algebra of dimension $r^2$ over $K$ with an involution $\sigma$ of second kind (i.e. $\sigma$ acts non-trivially ...
11
votes
1answer
496 views

Counting representations of $k[x,y]$ when $k$ is finite

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial ...
20
votes
4answers
1k views

Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$ a generalized polynomial if for any distinct integers $m$ and $n$ we have $(m - n)|(f(m)-f(n))$. It is easy to check that polynomial ...
2
votes
1answer
151 views

for which truth-operations f can f-membership in a prime ideal be represented by a polynomial?

Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital. Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's ...
2
votes
4answers
453 views

Products of Boolean algebras and probability measures thereon

These are really two questions, but the second presupposes the first. First, let $( B_i )_{i\in I}$ be an arbitrary family of Boolean algebras. I want to directly form a product of them that is like ...
3
votes
2answers
165 views

Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
11
votes
7answers
684 views

Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...
0
votes
0answers
16 views

Projecting solutions of Hermitian forms over local rings

Let $R$ be a local ring (commutative and with $1$) with maximal ideal $M$, with an involution $\theta$. Let $h$ be a Hermitian form on $R^n$, i.e. $h:R^n\times R^n\rightarrow R$ such that $h$ is ...
3
votes
3answers
942 views

a quotient of an injective module which is not injective

If R is an infinite direct product of fields, then R is an injective R-module...But I need an example of a quotient ring R, R/S, that is not injective ?? I feel that "if we take S as an infinite ...
5
votes
2answers
185 views

Matrix representation of real *-algebras

It is a standard fact that everyt real $n$-dimensional algebra is a subalgebra of $M_n(\mathbb R)$. The transposition map, operating in $M_n(\mathbb R)$, is an involutive ($(A^t)^t=A$) ...
14
votes
6answers
9k views

Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional. Is this true in general for ...
3
votes
1answer
64 views

Intersection of Maximal Left Ideals with Finite Dimensional Quotient

Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I ...
21
votes
6answers
810 views

How to recognize a Hopf algebra?

Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra? I guess in ...
5
votes
2answers
467 views

integral equivalence classes of quadratic forms

Let $A$ and $B$ two symmetric matrices definite positive over $\mathbb{R}$. Then we say that $A$ and $B$ are integrally equivalent if there exists $Q\in GL_n(\mathbb{Z})$ such that $A=Q.B.Q^t$ (1) ...
4
votes
0answers
134 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 ...
16
votes
0answers
184 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 ...
2
votes
1answer
989 views

Is there an algebraic proof of the infinitude of primes? [closed]

It is well-known that there exists a (justly celebrated) topological proof of the infinitude of primes (Hillel Fürstenburg, 1955). Does there also exist an algebraic proof?
12
votes
4answers
757 views

Showing that a family of polynomials has positive and real roots.

Hi everybody, for my research I am dealing with the following function: $$\alpha_n(x):=\left.\frac{\partial^{2n+1}}{\partial z^{2n+1}}\frac{\sinh(z)}{\cosh(z)-1+x}\right|_{z=0},\quad n\in ...
2
votes
1answer
118 views

Vertices of a polytope as algebra generators

I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...
0
votes
2answers
79 views

Finitely generated projective = finitely presented flat over a noncommutative Noetherian ring

Let $R$ be a possibly noncommutative left Noetherian ring and $M$ an $R$-module. I am looking for a reference or a proof for the following fact: $M$ is finitely generated and projective if and only if ...
9
votes
2answers
1k views

When the determinant of a 2x2 polynomial matrix is a square?

Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...
11
votes
2answers
617 views

Are there only finitely many associative algebras of fixed dimension?

Given an algebraically closed field $F$, for any positive integer $n$, are there always only finitely many non-isomorphic (noncommutative) associative algebras (possibly without identity) with ...
2
votes
1answer
271 views

How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...
1
vote
1answer
142 views

Universal constructions that factor through endomorphisms

If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor ...
2
votes
0answers
125 views

C*-algebras and bounded relations

I'm trying to get used to the language of generators and relations for C*-algebras through Loring's "Lifting Solutions to Perturbing Problems in C*-Algebras". So far this is what I got from the first ...
5
votes
1answer
167 views

Injective flat module

Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such. ...
10
votes
4answers
446 views

Morita equivalence for *-algebras

This is a reference request. I'm looking for a definition of Morita equivalence of *-algebras, as described below. If anyone thinks that this is not the right way to define Morita equivalence of ...
1
vote
0answers
206 views

Global dimension of a subalgebra with all units

(All rings here are always assumed to be unital and associative). Setup Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying: If $u$ is a unit ...
11
votes
2answers
1k views

Dimension of infinite product of vector spaces

This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring. It is well-known that an infinite dimensional vector space is ...
3
votes
0answers
99 views

Finding an integral basis for the lattice of the form $\mathbb{Z}^J \cap \mathbf{p}^{\perp}$

Let $\mathbf{p}$ be a primitive point in the lattice $\mathbb{Z}^J$ and denote the $J-1$ dimensional vector space $V = \mathbf{p}^{\perp} \subseteq \mathbb{R}^J$. Let $\Lambda' = \mathbb{Z}^J \cap V ...
3
votes
0answers
62 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, ...
0
votes
0answers
69 views

Grading a module

Let $R = \bigoplus_{n\in\mathbb{Z}}R_n$ be a graded ring. I'm trying to understand the structure of simple graded $R$-modules. In C. Nastasescu and F. Van Oystaeyen book, Methods of graded rings, ...
1
vote
1answer
98 views

Modules “projective in a subcategory”

In my research I have come up with the following notion which I would like to learn more about. It may be very naive. Let $R$ be a ring, $M$ an $R$-module and $S$ a class or $R$-modules closed under ...
4
votes
2answers
145 views

Is there a purely module theoretic characterization of semiprimitive rings?

A ring (say unital for simplicity) is semiprimitive (or Jacobson semisimple) if its Jacobson radical is trivial, or equivalently it has faithful semisimple module. Semiprimitivity is a Morita ...
3
votes
1answer
148 views

On the socle of rings

Is it possible that the socle of a ring (with identity) is cyclic as a left ideal but not finitely generated as a right ideal !?
3
votes
3answers
819 views

Defining Multiplication in Polynomials over Rings of Matrices

More explicitly, if $M_{2 \times 2}(\mathbb{R})[x]$ denotes the ring of polynomials over the ring of 2x2 matrices with real coefficients (with indeterminate x a 2 by 2 matrix with real coefficients), ...
3
votes
2answers
312 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
9
votes
3answers
1k views

What are hypergroups and hyperrings good for?

I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a ...
17
votes
3answers
1k views

Example for column rank $\neq$ row rank

The proof that column rank = row rank for matrices over a field relies on the fact that the elements of a field commute. I'm looking for an easy example of a matrix over a ring for which column rank ...
3
votes
1answer
117 views

What do epimorphisms in noncommutative rings look like?

The question I want to ask is inspired by this mathoverflow post about epimorphisms in the category of commutative rings. I found the seminar (by P. Samuel) referenced by David Rydh particularly ...
11
votes
3answers
2k views

Zero divisor conjecture for finite fields

I believe that the most attractive "zero-divisor" conjecture is the existence of non-trivial zero-divisors in a group ring $\mathbb{C}[G]$ for a torsion free group $G$. For the sake of knowledge let ...