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

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2
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420 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
2answers
506 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
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1answer
124 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
1answer
244 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
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2answers
251 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
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1answer
105 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
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1answer
374 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
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0answers
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 ...
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0answers
115 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
2answers
456 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
1answer
144 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
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1answer
96 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
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1answer
318 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
2answers
113 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
1answer
158 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
0answers
61 views

derivative of the logarithm of a complete homogeneous polynomials

I have the following complete homogeneous polynomial of degree $r$: $p_(x_1, x_2,...x_n) = \sum_{i_1 + i_2 + ... +i_n = r, i_k\in {0,1,..r}} \phi_{i_1}(x_2)\phi_{i_2}(x_2)...\phi_{i_n}(x_n) $ where ...
0
votes
1answer
245 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
1answer
214 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
2answers
472 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
2answers
262 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
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0answers
113 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
1answer
189 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
1answer
118 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
1answer
95 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
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1answer
355 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 ...
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0answers
73 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
2answers
107 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
1answer
141 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
2answers
404 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
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0answers
103 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
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1answer
185 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
1answer
120 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 ...
2
votes
1answer
162 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
1answer
189 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
2answers
439 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
1answer
128 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 ...
4
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0answers
191 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
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0answers
499 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
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0answers
221 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)) ...
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3answers
936 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 ...
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0answers
170 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
1answer
196 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 ...
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5answers
612 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: ...
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3answers
763 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 ...
1
vote
1answer
134 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 ...
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1answer
232 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 ...
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0answers
115 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 ...
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0answers
107 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 ...
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4answers
510 views

zeros of a homogeneous polynomial

Hi All, Let $F$ be a finite field, $\lambda\in F$, and $$p_\lambda (x,y,z)=\left|\begin{array}{ccc}x & y & z \\ y & z & x +\lambda z \\ z& x+\lambda z & y+\lambda x+\lambda ...
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1answer
161 views

Find a special element in group algebra

Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the ...