**0**

votes

**0**answers

4 views

### For which category (if any) are Lie algebras the algebras of a monad? [migrated]

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...

**4**

votes

**1**answer

85 views

### Left- and right-sided principal ideals of quaternions have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...

**1**

vote

**1**answer

131 views

### Polynomial ring operations on $\mathbb{Z}$

I have asked this on Math Stack Exchange but without answers:
The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables):
...

**5**

votes

**1**answer

141 views

### Bimodules of fractions

This question is about extending modules of fractions to bimodules of fractions. I would not be surprised if the result is known, I have tried looking in Goodearl and Warfield, but may have missed the ...

**4**

votes

**0**answers

135 views

### Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?

A fusion ring is a finite dimensional complex space
$\mathbb{C}\mathcal{B}$ together with a distinguished basis
$\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j =
\sum_k n_{ij}^kh_k $...

**3**

votes

**1**answer

32 views

### Normed algebras of indefinite signature?

Hurwitz's theorem states that a real possibly-non-associative algebra (meaning just a real vector space $V$ equipped with a map $m: V \otimes V \to V$) along with a positive definite quadratic form $|\...

**2**

votes

**0**answers

190 views

### Unique product groups (and semigroups)

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...

**0**

votes

**1**answer

113 views

### Definition of a cosemisimple Hopf algebra

A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the ...

**3**

votes

**0**answers

62 views

### Antipode action on quantum minors

Let ${\cal O}(SU_q(n))$ be the standard $q$-deformed coordinate algebra of $SU(n)$, with the canonical generators $x_{i,j}$. For $I = \{i_1,\ldots, i_r\}, J=\{j_1,\ldots,j_r\}\subseteq \{1, \ldots,n\}$...

**2**

votes

**0**answers

166 views

### Is there an integral simple fusion ring of multiplicity one and Frobenius type? (obvious excepted)

To avoid any confusion, we rewrite the basic definitions for a fusion ring (already written in this post).
A fusion ring is a finite dimensional complex space
$\mathbb{C}\mathcal{B}$ together ...

**2**

votes

**0**answers

114 views

### Torsion ideal in symmetric algebra

Let $D$ be a commutative domain, $M$ a $D$-module without torsion, and $S(M)$ its symmetric algebra.
Is the $D$-torsion ideal of $S(M)$ the prime ideal of $S(M)$?

**1**

vote

**1**answer

38 views

### On finite Uniform (Goldie) dimensions

1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions?
2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to $M$...

**2**

votes

**0**answers

112 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

**0**answers

151 views

### For $f: X \to (Y \to X)$, what is the name of the property whereby for all $x\in X$ and $y_1, y_2 \in Y,$ $f(f(x,y_1), y_2) = f(f(x, y_2),y_1)$?

My PL group has been discussing this (so we are really professional reject mathematicians, but this is more about the math behind what we are doing).
Some of us called it a "generalized associativity,...

**2**

votes

**1**answer

139 views

### Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings
$$
\begin{array}{}
R & \xrightarrow{f_2} & R_2 \\
\downarrow{f_1} & &...

**5**

votes

**1**answer

153 views

### Is there a way to embed Clifford algebras into the corresponding tensor algebra?

$\newcommand{\talg}{\mathcal{T}(V)}$$\newcommand{\clalg}{\mathcal{Cl}_q(V)}$$\newcommand{\qalg}{\mathcal{I}_q(V)}$Is there a way to embed Clifford algebras into the corresponding tensor algebra?
...

**3**

votes

**2**answers

114 views

### Dimension of preprojective algebra of Dynkin type

Fix a field $\Bbbk$. Let $Q$ be a Dynkin quiver and let $\Pi(Q)$ be its preprojective algebra. It is well-known that in this case $\Pi(Q)$ is finite-dimensional, but I've been unable to find a ...

**0**

votes

**1**answer

46 views

### What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$.
What are ...

**13**

votes

**2**answers

650 views

### Bass' stable range condition for principal ideal domains

In his algebraic K-Theory book Bass gives the following property on a ring $R$ and a number $n$:
For every $n$ elements $v_1, \ldots, v_n$ that generate the unit ideal there are numbers $r_1, \ldots ...

**57**

votes

**2**answers

5k views

### How would you solve this tantalizing Halmos problem?

$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
...

**2**

votes

**1**answer

158 views

### Annihilator of minimal prime ideal in a commutative Noetherian ring

Let $M$ be an $R$-module of finite length and $N$ a maximal submodule of $M.$
Is there an element $m$ in $M$ such that $m(N:M)=0$?
It is a generalization of this result:
In a Notherian ring $...

**1**

vote

**1**answer

98 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

41 views

### 4-D lattices and quaternion

It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ...

**43**

votes

**19**answers

39k views

### Linear Algebra Texts?

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...

**2**

votes

**0**answers

134 views

### Inverse limit of Noetherian rings

Let $R_i$ be a noetherian regular local ring of Krull dimension being finite. Suppose we are given a surjective homomorphism $\phi_{i,j} \colon R_{j} \twoheadrightarrow R_i$ for each $i,j$ with $j >...

**0**

votes

**1**answer

97 views

### the relation between projective and quasi-projective modules

An $R$-module $M$ is called quasi-projective if $\text{Hom}_R(M,M)\to\text{Hom}_R(M,N)$ is surjective for every surjective homomorphism $M\twoheadrightarrow N$.
What are the rings $R$ for which every ...

**2**

votes

**1**answer

143 views

### Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map
$$
\pi:SL_n(R)\rightarrow SL_n(R/I)
$$
(In the original question I had put $GL_n$ instead of $SL_n$ ...

**1**

vote

**1**answer

262 views

### Does a nonabelian Picard group exist?

Over a noncommutative algebra $A$ we have no problem in defining invertible bimodules (as in the book by Bass on algebraic $K$-theory) - corresponding to line bundles over topological spaces $X$ if $A=...

**1**

vote

**0**answers

97 views

### depth of ideal in polynomial ring

Let $R$ be a Noetherian local ring. Then the "depth" of an ideal $I$ measures the maximal length of regular sequence inside $I$. And $depth (R/I)$ measures the maximal length of regular sequence in ...

**7**

votes

**0**answers

241 views

### How bad can $SK_1$ of a commutative ring be?

For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...

**4**

votes

**1**answer

98 views

### When is the category of Gorenstein projective $R$-modules Frobenius?

Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a ...

**12**

votes

**4**answers

1k views

### For which rings R is SL_n(R) generated by transvections?

Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
Under which conditions is the group $SL_n(R)$ generated by transvections?
(A transvection is a matrix with $1$ everywhere ...

**5**

votes

**1**answer

114 views

### Integer Gelfand-Kirillov dimension

Let $R$ be a (noncommutative) Noetherian affine $K$-algebra. The Gelfand-Kirillov dimension is known to be an integer for many classes of affine Noetherian algebras. I wonder, if this is true for any ...

**1**

vote

**0**answers

74 views

### Question about radical ideal

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1,...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], \deg(f) \...

**0**

votes

**0**answers

814 views

### Does anyone know the definition for Hochschild cohomology of a differential graded algebra?

Let $A$ with $d$ be a differential graded associative algebra ($DG$ algebra).
What is the definition of the Hochschild complex $C^*(A,A)$?
There should be a Hoshschild differential and a cup ...

**5**

votes

**0**answers

101 views

### “Factorisation” in special linear groups over rings of integers

It is known that for any number field $F$ with infinitely many units (i.e. $F$ is not $\mathbb Q$ or an imaginary quadratic field) with ring of integers $O$ the special linear group $\mathrm{SL}_2(O)$ ...

**2**

votes

**0**answers

110 views

### Parametric surfaces in $\mathbb{R}^4$ via quaternion multiplication of curves

In this question, $\mathbb{R}^4$ is identified with the quaternions $\mathbb{H}$ via the map $$(a_1, a_2, a_3, a_4) \mapsto a_1+ia_2+ja_3+ka_4,$$
where $1,i,j,k$ are the standard basis elements of ...

**0**

votes

**0**answers

237 views

### Isomorphic maximal commutative semi-simple sub algebras of complex matrices

When giving $A_1,A_2$ two isomorphic maximal commutative semi-simple sub algebras of $M_n(\mathbb{C})$, are these algebras conjugate in $M_n(\mathbb{C})$? Namely, does there exists a regular matrix $P$...

**2**

votes

**1**answer

213 views

### Cyclic modules over local rings

Let $R$ be a local ring. If two cyclic right $R$-modules are epimorphic images of each other, are these modules necessarily isomorphic?

**8**

votes

**1**answer

273 views

### When is bar-cobar duality an equivalence?

Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...

**6**

votes

**1**answer

234 views

### coefficient-wise powers of matrices. Reference wanted

Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...

**1**

vote

**1**answer

89 views

### universal enveloping algebra of Leibniz algebra

According to Loday and Prashvili's paper, to defined universal enveloping algebra of Leibniz algebras we need $g^{r}$ and $g^{l}$ as two copies of the Leibniz algebra $g$. What does it mean by ...

**1**

vote

**1**answer

74 views

### Derivations of polynomial identity rings

Does anyone have any references to general theory of derivations of PI rings? I have had a quick look around without much luck.

**6**

votes

**1**answer

123 views

### When does a quadratic algebra have zero divisors?

Consider two quadratic algebras based on the vector space $\mathbb{R}^3$ with basis $x,y,z$. The antisymmetric tensor algebra $\Lambda \mathbb{R}^3$ obviously has zero divisors, e.g. $(x)(xy)=0$, but ...

**3**

votes

**1**answer

183 views

### Geometric contractibility of noetherian rings

Let $A$ be a noetherian ring. Let us define $A$ to be $n$-contractible if
All locally free sheaves of rank $\le n$ over $\text{Spec} A$ is trivial.
There exist a non-trivial locally free sheaf of ...

**1**

vote

**0**answers

126 views

### What inner autmorphism really do in abstract algebra? [closed]

The definition of inner automorphism is easy to understand but I wonder what these inner automorphism really do in abstract Lie algebra, especially when we are talking about Lie algebra ?
In Lie ...

**12**

votes

**3**answers

888 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 ...

**4**

votes

**0**answers

126 views

### Weyl algebra acting on a polynomial ring

Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots,
x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl
algebra. As usual $W$ ...

**6**

votes

**0**answers

168 views

### Is there an explicit construction of Tate-Beilinson residue?

Background: Tate's construction of abstract residues is generalized in Beilinson's constructon as follows:
Let $E$ be a unital, cubically decomposed $k$-algebra, i.e., (a) there are two sided ideals $...

**12**

votes

**0**answers

170 views

### Concept associated to the Eudoxus reals

I am aware of three different constructions of the field of real numbers :
The Cauchy sequence construction : in this case, we see the field $\mathbb{Q}$ as a metric space and $\mathbb{R}$ is the ...