**3**

votes

**1**answer

165 views

### Is the definition of Gerstenhaber bracket related to operads?

I was reading these notes by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by
$$
...

**8**

votes

**1**answer

336 views

### Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...

**1**

vote

**1**answer

100 views

### Invariance of reduced trace of Azumaya algebras

Let $A$ be an Azumaya algebra over the commutative ring $R$ and let
$\newcommand{\tr}{\operatorname{tr}} \tr\colon A \to R$ be the reduced trace as defined (for example) in Section IV.2 of M.-A. Knus, ...

**2**

votes

**1**answer

458 views

### Commutative associative rational binary operations

What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of ...

**4**

votes

**2**answers

536 views

### Rings with a finite maximal ideal

A field has a finite maximal ideal. Of course the converse is not true. So is there a characterization for those rings having a finite maximal ideal ??
EDIT: The characterization below is very nice. ...

**1**

vote

**1**answer

130 views

### Idempotents in Green J classes

I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular:
A $\mathcal J$-class containing an idempotent is called regular. ...

**4**

votes

**1**answer

270 views

### Rings with group of units cyclic of prime order

For what prime numbers $p$ there exists a ring with identity and exactly $p$ invertible elements ?
REMARK It can be shown that for $p=5$ there is no such ring, so I am wondering for what values of ...

**3**

votes

**2**answers

258 views

### Do morphisms of finitely-decomposable Quiver representations map indecomposables nicely?

Consider two quivers $Q$ and $Q'$ of type $A_n$, laid out horizontally like so:
Given representations of $Q$ and $Q'$, Gabriel's theorem guarantees the existence of finitely many indecomposables ...

**2**

votes

**1**answer

113 views

### A non-orthomodular orthocomplemented lattice identity?

Assume you have an orthocomplemented (but possibly not orthomodular) lattice $L$. For $q,r\in L$ say "$q$ and $r$ are in position $p'$" to mean that $q\wedge r^\perp=0$ and $r\wedge q^\perp=0$. Is ...

**4**

votes

**1**answer

398 views

### “as close to being semisimple as it can possibly be.”

I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here.
In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality ...

**2**

votes

**0**answers

79 views

### Usefulness of polynomial ideals for graphs

Given a graph $G=(V,E)$ on $n$ vertices, one can associate to it the polynomial $f \in k[x_1,\dots,x_n]$ given by $f = \prod_{\{i,j\}\in E}(x_i - x_j)$. Then one can map various graph properties into ...

**1**

vote

**0**answers

130 views

### simple tensor product of modules over algebras

Let $M$, $N$ be simple modules over associative algebras $A$ and $B$ (over $\mathbb{C}$), respectively. When is $M\otimes N$ simple as a $A\otimes B$-module?
It is right if $A$ or $B$ has a ...

**3**

votes

**2**answers

482 views

### Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...

**7**

votes

**1**answer

163 views

### Two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers

I am searching for two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers. Who knows of such examples? Thanks a lot.

**2**

votes

**1**answer

99 views

### Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...

**11**

votes

**1**answer

323 views

### $\mathbb{Z}G$ (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 ...

**2**

votes

**2**answers

124 views

### Are there atomistic ortholattices which are not modular?

Let $L$ be an atomic ortholattice. We say that two elements $a$ and $b$ of $L$ are orthogonal if $a\leq b^\perp$. If $L$ is orthomodular then every element of $L$ can be written as a join of pairwise ...

**7**

votes

**1**answer

168 views

### $K_0$ group of graph underlying an approximately finite (AF) C* algebra

Say we have an AF C* algebra $A$ described by some Bratteli diagram $E$. If $M_\infty (A)=\displaystyle{\lim_\rightarrow M_n(A)}$ and $P(A)$ are the projections in this algebra, we know that ...

**0**

votes

**1**answer

267 views

### GCD computation for multiple polynomials and degree of Bezout coefficients

Assuming two polynomials $P_1,P_2 \in \mathbb{Z}_p[r]$ of degree $n$, with no common factors, we know that there exist polynomials $Q_1,Q_2$ s.t.: $Q_1P_1 + Q_2P_2 =1$. From Bezout's identity we also ...

**0**

votes

**1**answer

225 views

### Finite extension of a field [closed]

Is it true, that if $A$ is finitely generated commutatative algebra over a field $k$, not necessary algebraically closed, then prime ideal $p \subset A$ is maximal if and only if $k \subset Quot(A/p)$ ...

**4**

votes

**2**answers

524 views

### Example of a reduced ring whose completion is not reduced

Let $(R,\mathfrak{m})$ be a local Noetherian ring. Give an example such that R is reduced but $\mathfrak{m}$-adic completion of R is not reduced.

**7**

votes

**2**answers

274 views

### Where should I search for resolutions?

In my research, to test some conjectures or just to illustrate some facts, I often need to compute some explicit examples of derived functors (in the sence of Quillen's model categories). Mainly I ...

**1**

vote

**0**answers

122 views

### Ring of Witt Vectors and Tensor product of Fields

Let $p > 2$ be a prime, and let $\textbf{F}_{p} =
\textbf{Z}/p\textbf{Z}$. Let $k_{1}$ be a finite field over
$\textbf{F}_{p}$, and let $k$ be a perfect field of characteristic
$p$. Then we have ...

**3**

votes

**1**answer

195 views

### Polynomial identities for mod p matrices

Can there be a polynomial over the field $F_p$ of $p$ elements ($p$ prime) in non-commuting variables $X_1,..., X_r$ such that:
1) $f(A_1,...,A_r)=0$ for every $n \times n$ matrices $A_1,...,A_r$ ...

**3**

votes

**1**answer

120 views

### something like a kleene algebra of rational functions?

There's a procedure that control engineers (used to) do to calculate the transfer function of a linearized system, gradually reducing a block diagram to a rational function of s. It's justified by ...

**1**

vote

**1**answer

108 views

### cobar construction of a direct sum of coalgebras

I have the following question. Let's denote by $\Omega$ the cobar functor, by $C$ some DG-coalgebra (not co-commutative) over a field $k$. Also suppose $V=k\times\dots\times k$ is just the product of ...

**14**

votes

**1**answer

379 views

### 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 ...

**2**

votes

**0**answers

83 views

### Nontrivial examples of rings of relative stable rank 1

Given a (commutative) ring $R$, one says that the stable rank $sr(R)\leq n$ if any unimodular row $(a_1,\ldots,a_{n+1})$ of length $n+1$ is stable, i.e. there exist $x_1,\ldots,x_n$ such that ...

**2**

votes

**2**answers

108 views

### Specialization of PBW-algebras over rational function field

I have been thinking about the concept of specialization of algebras defined over the field of rational functions $k = \mathbb{C}(t)$ (I'm using $\mathbb{C}$ for my work, but the question can be asked ...

**3**

votes

**1**answer

147 views

### Morphism of algebras

Let $Q(i)$ be the extension of the rational numbers $Q$ obtained by adjoining a root i of the polynomial $X^2 + 1$.
Consider the algebra B defined by the Hilbert symbol $(-2, -5)$ over $Q(i)$. So, by ...

**11**

votes

**2**answers

420 views

### What is a Kelley ring?

I've heard that in some book by someone named Kelley, perhaps an early edition of John L. Kelley's General Topology, the author gave a definition of a ring which turned out to be weaker than the usual ...

**4**

votes

**0**answers

104 views

### What is the composition in SesquiAlg?

To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and ...

**4**

votes

**1**answer

186 views

### Fermat’s Two Squares for polynomials

Is there an analog of Fermat’s Two Squares theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?
Fermat two squares says primes $p$ is sum of two squares ...

**1**

vote

**1**answer

122 views

### Finding reducible polynomials with restricted factors

Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with ...

**3**

votes

**1**answer

186 views

### Existence of a generalized matrix inverse over an arbitrary field?

Let $A\in M_n(K)$ be a square matrix over a field $K$. The notion of inverse matrix was generalized by Moore and Penrose for real and complex matrices
(also called pseudo-inverse $A^{\dagger}$ of $A$, ...

**4**

votes

**1**answer

198 views

### A question about Weil algebra

Let $G$ ba a compact Lie group with Lie algebra $\mathfrak{g}$, $\mathfrak{g}^{*}$ be the dual of $\mathfrak{g}$. We known the Weil algebra is
$$W(\mathfrak{g})=\wedge(\mathfrak{g}^{*})\otimes ...

**7**

votes

**2**answers

544 views

### Quintic polynomial solution by Jacobi Theta function.

Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at ...

**3**

votes

**1**answer

132 views

### Thinking about the quadratic dual graphically

Say you have a quadratic algebra $A$ , that is, an algebra defined by a finite list of generators over a ground ring $k$ (either a field or a direct product of fields, which will be assumed ...

**5**

votes

**0**answers

198 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 ...

**12**

votes

**0**answers

514 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 ...

**2**

votes

**0**answers

260 views

### Errata For Bourbaki Algebra Chapters 1 - 3

I am trying to teach myself some algebra by reading Bourbaki's Algebra (en, 2nd printing, 1998). Reading through Chapter 2, §1, I find that there are a couple of mistakes.
(no. 2, paragraph (-7)) ...

**20**

votes

**4**answers

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 ...

**5**

votes

**0**answers

185 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 ...

**0**

votes

**1**answer

207 views

### Difficulty understanding Russian- Paper in Galois Theory

While searching for some useful results in Galois theory, I encountered a paper of Shafarevich from 1956 .
Its first page is here:
http://i44.tinypic.com/2emfomw.jpg
...

**18**

votes

**5**answers

651 views

### Bass' stable range of $\mathbf Z[X]$

Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication:
...

**18**

votes

**3**answers

817 views

### Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ?

In general, it seems not known which finite abelian groups are class groups of quadratic number fields.
For imaginary quadratic number fileds, I read that $(\mathbb{Z}/3\mathbb{Z})^3$ is the smallest ...

**2**

votes

**1**answer

162 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 ...

**1**

vote

**1**answer

239 views

### Are the elements of a division algebra which commute with all commutators in the center of the algebra?

I asked this quetion five days ago at http://math.stackexchange.com/questions/406669/are-the-elements-of-a-division-algebra-which-commute-with-all-commutators-in-the Some good people have given good ...

**1**

vote

**0**answers

119 views

### Rings with the property $\dim R-\dim R/p\leq \text{const}$ for all minimal $p$

I'm curious if there exists a class of rings generalizing quasi-unmixed rings. I guess a generalization of quasi-unmixed rings can be done as follows:
For a fixed integer $i$
$$\forall ...

**2**

votes

**0**answers

112 views

### when upper triangular matrix modulo prime ideals implies upper triangular?

Let $E/ \mathbb{Q}_p$ be a finite extension, let $\mathcal{O}$ be the ring of integers of $E$. Let $A$ be a reduced noetherian local complete $\mathcal{O}$-algebra with the maximal ideal ...