**0**

votes

**0**answers

37 views

### Homomorphism from integral module generated by roots of unity to cyclic group?

Let $S$ be the set of all roots of unity. Consider the $\mathbb{Z}$-module, $\mathbb{Z} S$, as an additive abelian group (that is, $\mathbb{Z} S$ is the subset of complex numbers that can be ...

**4**

votes

**0**answers

92 views

### Whether a given algebra is the algebra of endomorphisms for a vector space

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...

**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

**2**answers

216 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):
...

**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 ...

**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 $|\...

**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 $I = \{i_1,\ldots, i_r\}, J=\{j_1,\ldots,j_r\}\subseteq \{1, \ldots,n\}$...

**5**

votes

**1**answer

143 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 ...

**2**

votes

**0**answers

173 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 ...

**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 $...

**5**

votes

**1**answer

156 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?
...

**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 ...

**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 ...

**2**

votes

**0**answers

135 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 $SL_n$ ...

**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} & &...

**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

99 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 ...

**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 ...

**5**

votes

**0**answers

104 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 $\mathrm{SL}_2(O)$ ...

**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 ...

**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?

**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 ...

**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 ...

**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 ...

**4**

votes

**0**answers

127 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 $...

**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) \...

**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 ...

**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

33 views

### Notion of trace in a Jordan algebra

Let A be a Jordan algebra (with identity). If x is in A let A[x] be the subalgebra generated by x and the identity. An element x is regular if the dimension of A[x] is maximal.
For x regular, denote ...

**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

**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.

**2**

votes

**2**answers

234 views

### Attaching an ideal whose square is zero: does this operation have a name and a notation?

I know I met the following construction somewhere, but I cannot remember where. Let $A$ be
a (unital associative) ring, and let $N$ be an $A$-$A$ bimodule. On the product set $A\times N$ we define ...

**1**

vote

**0**answers

39 views

### Computing intersection of Weyl algebra ideal with certain subring

Let $D=k [x_1,\ldots, x_n, \partial_1,\ldots, \partial_n] $ be the nth Weyl algebra over the characteristic zero field $k $. Set $\theta_i=x_i\partial_i $. Let $I $ be a left ideal in $D $. Is there a ...

**4**

votes

**2**answers

501 views

### Some examples of clean topological spaces

I asked this question at MSE but I did not received any answer, so I repeat it here at MO:
What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X)$...

**9**

votes

**1**answer

131 views

### Can a semigroup with zero be globally isomorphic to a semigroup without zero?

This is not a great question for sure and it may even be trivial for all I know, but a couple of years ago, when I still thought I'd be a mathematician, I spent quite a lot of time thinking about it ...

**6**

votes

**1**answer

235 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,...

**3**

votes

**0**answers

55 views

### Reference for classification of positive involutions

An involution on a finite dimensional algebra $A$ over $\mathbb{Q}$ is an involutive anti-automorphism of $A$. If $\sigma$ is an involution on $A$, we say that $\sigma$ is positive if $\mathrm{Tr}_{A/\...

**2**

votes

**0**answers

66 views

### GK dimension of generalized Weyl algebras

I believe that the GK dimension for a generalized Weyl algebra $D(\sigma,a)$ is just the GKdim$(D) + 1$.
Does anyone have a reference for this?
I can find partial results, and I am sure this is ...

**5**

votes

**1**answer

97 views

### Why is $\operatorname{nr}_{F[G]}:K_1(F[G])\to Z(F[G])^\times$ a bijection?

Let $A$ be a finite dimensional semisimple $F$-algebra and $K_1(A)$ the Whitehead group of $A$.
By splitting $A$ into its Wedderburn components, the reduced norm map $\operatorname{nr}_A:K_1(A)\to Z(...

**5**

votes

**0**answers

149 views

### If $A$ is an algebra, $Sym^n(A)$ is an algebra. Where can I learn more about this algebra structure?

$\newcommand{\Vect}{\mathsf{Vect}}
\newcommand{\nats}{\mathbb{N}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\Alg}{\mathsf{Alg}}
\newcommand{\CAlg}{\mathsf{CAlg}}
\newcommand{\Hom}{\mathrm{Hom}}$
Let ...

**15**

votes

**1**answer

538 views

### Swan K-theory of Z/4

Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with ...

**1**

vote

**0**answers

147 views

### What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...

**4**

votes

**0**answers

270 views

### Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...

**3**

votes

**0**answers

148 views

### For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]:
If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free $...

**4**

votes

**1**answer

180 views

### Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...

**1**

vote

**1**answer

163 views

### Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...