**15**

votes

**1**answer

496 views

### How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, … algebras?

There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique.
What about just traces on separate algebras? That is, take one of them,...

**2**

votes

**0**answers

52 views

### Locally free sheaves of algebras vs. algebra bundles

It is well known that there is a bijective correspondence between locally free sheaves of modules and vector bundles (cf. https://rigtriv.wordpress.com/2008/04/09/locally-free-sheaves-and-vector-...

**4**

votes

**2**answers

313 views

### Relation between Associative algebra and group algebra

Let $A$ be an associative algebra over a filed $k$.
Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$?
I am ...

**1**

vote

**2**answers

177 views

### Degree of sum of integral elements over a UFD

Is it possible to generalize Degree of sum of algebraic numbers (especially Pete L. Clark's answer, based on Keith Conrad's answer)
in the following way:
Let $D$ be a (noetherian) UFD of zero ...

**1**

vote

**0**answers

98 views

### Programmatically computing dual Hopf algebras: state of the art

Given a graded Hopf algebra of finite type, we know the (graded) linear dual is also a graded Hopf algebra. For instance the dual Hopf algebra to the polynomial algebra on an even degree generator, $R[...

**9**

votes

**0**answers

137 views

### When does Hochschild homology commute with infinite products?

Let $A$ be an associative algebra. Its zeroth Hochschild homology $\mathrm{HH}_0(A)$ is the cokernel of the linear map $A^{\wedge 2} \to A$, $a \wedge b \mapsto ab - ba$. I.e. you quotient the ...

**1**

vote

**1**answer

194 views

### a problem about ideals of polynomial rings

Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let $(...

**8**

votes

**2**answers

417 views

### Strategies for proving a category is Noetherian?

Let $C$ be a small linear category over a commutative ring $R$. A representation of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation $[...

**12**

votes

**3**answers

881 views

### The number of ideals in a ring

Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number ...

**0**

votes

**1**answer

68 views

### Is the following ideal zero-dimensional?

Let $\left\{ p_{k}\in\mathbb{R}\left[ x_{1},\ldots,x_{n-1}\right]
:k=1,\ldots,n\right\} $ be a family of linear polynomials such that
$p_{k}\left( 0,\ldots,0\right) =0$. Let $\alpha_{i},\beta_{i}\...

**6**

votes

**0**answers

161 views

### How to prove that a projective module is not free?

Let $A$ be a noncommutative (perhaps $\ast$-) algebra (over $\mathbb{C}$) and let $M$ be a projective module defined via a projector $P\in M_n(A)$; i.e. $M=P(A^n)$. Furthermore, assume that all ...

**1**

vote

**1**answer

355 views

### ideals of polynomial ring of two variables generated by two elements

Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by
$$
f(x,y)=x^4-3xy+y^2,$$
$$
g(x,y)=x^5-4xy+3xy^2.$$
Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$.
Is $x,x^2,...

**0**

votes

**0**answers

115 views

### Action of semidirect products of cyclic groups

Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is ...

**1**

vote

**0**answers

86 views

### Global dimension of graded Lie algebra

The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm Ext}^...

**7**

votes

**2**answers

760 views

### How do I apply the Boolean Prime Ideal Theorem?

I have become aware of an amazing phenomenon from a myriad of questions and answers here on MathOverflow: many of the results that I would typically prove using the Axiom of Choice can actually be ...

**4**

votes

**0**answers

58 views

### examples of local, nonsemisimple , nonsymmetric hopf algebras

I'm searching for (a class of) examples of Hopf algebras , which have the following properties:
they should be finite dimensional
they should not be semisimple
they should be local
they should ...

**7**

votes

**1**answer

199 views

### Noncommutative group of invertible ideals of a ring

Let $R$ be a noetherian domain and let $\mathcal{O}$ be an $R$-algebra that is finitely generated and projective as an $R$-module. The set of invertible fractional ideals of $\mathcal{O}$ is a group ...

**4**

votes

**1**answer

113 views

### Locally nilpotent operators of the Weyl algebra

$\newcommand{\ad}{\operatorname{ad}}$As my recent post (here) did not receive any answers yet, I thought I would ask a similar question in which I'm also interested.
Let $A=$ $^{k \langle x,y\rangle }...

**4**

votes

**0**answers

99 views

### Nilpotent operator of the Weyl algebra

For a research project I'm currently working on, I came across the following problem:
Let $A=$ $^{k <x,y> }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $p$, where $...

**1**

vote

**1**answer

144 views

### Structure of $\text{Aut}_R(R[X])$

Let $R$ be a commutative ring with identity. I'd like to know how to determine the set $\text{Aut}_R(R[X])$ of all $R$-automorphisms of $R[X]$.
I've proved that all $\sigma\in\text{Aut}_R(R[X])$ ...

**5**

votes

**1**answer

150 views

### If $R$ is generated by idempotents, then $\text{Ann}(R)=0$?

Let $R$ be a ring (not necessarily commutative or unital) that is generated by idempotents. I'd like to know if $\text{Ann}(R)=0$ must hold. Here I use $\text{Ann}(R)$ to denote the set of all ...

**2**

votes

**0**answers

103 views

### Geometry of rings and semi-rings

Basically, I wonder whether a theory similar to geometric group theory has been or could be developed for rings and semirings.
One direction would be the following. Consider $\mathbb{N}$ (with the ...

**1**

vote

**1**answer

143 views

### Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...

**2**

votes

**0**answers

51 views

### Goldie's Theorem for Semigroups

Goldie's theorem is a theorem in noncommutative ring theory that gives a clear picture of semiprime Noetherian rings (actually a slightly broader class). Let $R$ be a semiprime Noetherian ring. The ...

**2**

votes

**1**answer

148 views

### When does $\overline{U(0,1)}=B(0,1)$ hold?

Given $R$ an absolute valued ring (with unit), sometimes $\overline{U(0,1)}=B(0,1)$ (for example, $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{H}$) and sometimes $\overline{U(0,1)}\neq B(0,1)$ (for ...

**1**

vote

**1**answer

149 views

### Example of a left perfect ring with finite left global dimension that is not right coherent

I am looking for an example in noncommutative ring theory. Namely, I am looking for a left perfect ring with finite left global dimension that is not right coherent. It seems to me that should be ...

**0**

votes

**0**answers

146 views

### For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?

For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo $...

**8**

votes

**0**answers

220 views

### What's the analogue of a Young symmetrizer in the Brauer algebra?

According to Schur--Weyl duality, the centralizer of $\mathrm{GL}(V)$ acting diagonally on $V^{\otimes N}$ is the group algebra of the symmetric group $\mathbb S_N$. An equivalent formulation is the ...

**0**

votes

**1**answer

202 views

### Scalar restriction and scalar extension

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This yields the two functors $h_*\colon{\sf Mod}(S)\rightarrow{\sf Mod}(R)$ (scalar restriction) and $h^*\colon{\sf Mod}(R)\...

**2**

votes

**1**answer

547 views

### Automorphisms of Clifford Algebras

What are the automorphisms of real Clifford algebras $Cl_{n,0}$? Of course, I'm interested in the case where they are not central simple.

**0**

votes

**0**answers

133 views

### Rational group scheme

Suppose $G$ is a group scheme over a field $k$, i.e., $G$ is a functor from the category $\text{Alg}_k$ of unital commutative, associative $k$-algebras to the category of $\text{Groups}$. Suppose that ...

**1**

vote

**0**answers

63 views

### Convention on Clifford Product

When studying the Clifford Algebra associated to some $(V,Q)$, one finds two basic identities differing by a sign:
$vv=Q(v)$ (see, for instance, Wikipedia)
$vv=-Q(v)$ (see, for instance, MathWorld ...

**1**

vote

**0**answers

163 views

### local rings with finite type maximal ideal

Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal).
Is there a sufficient condition for $A$ to be noetherian?
For example, we know that the completion $\...

**0**

votes

**0**answers

41 views

### Embedding rational simple algebras in the real quaternions [duplicate]

Is there any way to embed a rational division algebra of dimension higher than 4 over its center in the real quaternions ?
I think not, but I cannot prove it.

**6**

votes

**0**answers

147 views

### flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...

**4**

votes

**1**answer

855 views

### Who coined “mob” and “clan” and why these words?

A mob is a word used for a topological semigroup which is a Hausdorff space. A clan is a compact connected mob with a two-sided identity element.
Who used these words with these meanings first and ...

**1**

vote

**1**answer

156 views

### configuration spaces of real projective space

Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring
$$
H^*(F(\mathbb{R}P^n,k);R)$$
is obtained for any ...

**2**

votes

**1**answer

155 views

### Involution on the components of a group algebra

If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has $(\...

**11**

votes

**0**answers

166 views

### Must nonunit in group algebra of free group generate proper two-sided ideal?

Let $F$ be a free group and $k$ be a field. If $x$ is an element of
the group algebra $k[F]$ that is not a unit (equivalently, that is not
a nonzero scalar multiple of an element of $F$), must the 2-...

**9**

votes

**0**answers

153 views

### Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...

**2**

votes

**1**answer

133 views

### Units in a finite semisimple group algebra

Let $G$ be a finite group and $k$ a finite field, with the characteristic of $k$ not dividing the order of $G$. Then $kG$ is a finite semisimple group algebra with the interesting property that an ...

**1**

vote

**0**answers

43 views

### About alternating polynomials an the Rowen's notation

Some definitions...
Definition 1: A polynomial $f(X_1,\dots ,X_d)$ is $t$-linear if the variables $X_1,\dots ,X_t,\; t\leq d$ appear in all monomials of $f$ and degree of $X_i,\; i=1,2,\dots ,t$ on ...

**2**

votes

**1**answer

184 views

### What are some good references on the Galois theory, factorization, or minimality of differential equations?

Let $\lambda$ be a nonzero complex number and let $u(x)$ be some smooth function $\mathbb{R}\to\mathbb{C}$, not identically zero. I want to prove that if $u$ satisfies $$u'' + \lambda u'+ \lambda^2 u =...

**1**

vote

**0**answers

96 views

### Equivalence of Kahler structures of based loop group and its Grassmannian model

In Pressley-Segal's Loop Groups, we have the following spaces equipped with Kahler structures. Let $G$ be a compact, connected, (simply connected) group with Lie algebra $\mathfrak g$.
Let $\...

**1**

vote

**0**answers

92 views

### Are the Standard Quantum Groups Coordinate Rings Noetherian?

Are the standard quantum groups $C_q[G]$ Noetherian and if so what is a standard reference?

**20**

votes

**2**answers

929 views

### What do you do if you believe a problem is undecidable?

While the title of this question is subjective, I hope to make what I'm looking for quite concrete. The first, and main question is this: If you believe that a problem you are working on is formally ...

**10**

votes

**2**answers

370 views

### Computing intersection of subrings

Let $R$ be a finitely generated commutative ring over a field, for concreteness.
If $S,T \leq R$ are two finitely generated subrings, is their intersection
also finitely generated?
(Certainly ...

**1**

vote

**0**answers

90 views

### Normalization (integral closure) of $\mathbb Z_p[x]$ in function field of a curve to obtain Model of curve

I want to follow this construction of a normal model of a curve:
Let $p\neq 2,3$ and $Y\to \mathbb P¹$ be a smooth projective curve over $\mathbb Q_p$ with function field $L/\mathbb Q_p(x)$ e.g. $L=\...

**3**

votes

**0**answers

210 views

### Certain conditions on cancellative semigroups

This is extracted from this question following Benjamin Steinberg's suggestion.
For a semigroup $S,$ let $P(S)$ denote the power semigroup of $S,$ which is made up of all non-empty subsets of $S$ ...

**6**

votes

**1**answer

292 views

### Rings satisfying the polynomial equation $x^4=x^2$

There are many results concerning the commutativity of rings satisfying a polynomial equation. I want to know if there is any result/reference about (finite) rings that satisfy the polynomial equation ...