**0**

votes

**0**answers

31 views

### Groebner basis of algebraic system of polynomials [on hold]

I have 8 polynomials with 8 unknowns as {p,L,x1,x2,y1,y2,z1,z2}, and the remaining are all known coefficients. The polynomials are as follows:
f1=h1*p - L*(h1*xb*y2 - h2*x1*y1 - h3*x2*y1) + h4;
...

**1**

vote

**0**answers

53 views

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

A curve $r(t)=(u(t), v(t), p(t), q(t)): \mathbb{R} \to \mathbb{R}^4$ can also be thought as a quaternion function $r(t)=u +i v + j p + kq$, where $1,i,j,k$ are the standard basis elements of the ...

**0**

votes

**0**answers

31 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 ...

**3**

votes

**0**answers

72 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

18 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

224 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

34 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

481 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 ...

**8**

votes

**1**answer

107 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 ...

**1**

vote

**0**answers

62 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 ...

**3**

votes

**0**answers

51 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 ...

**1**

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**0**answers

54 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 ...

**5**

votes

**0**answers

141 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

502 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

145 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

238 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

144 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

175 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

160 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.
...

**0**

votes

**2**answers

188 views

### Does there exist an Affinization or Projectivization process for Varieties?

Let us consider the classical isomorphism of real manifolds between $S^2$ and ${\mathbb CP}^1$. First strange thing we have here is that both are varieties, but $S^2$ is an affine and ${\mathbb CP}^1$ ...

**1**

vote

**2**answers

90 views

### Invertibility of all left multiplication maps in non-unital rings

Suppose that $R$ is a ring, not necessarily commutative nor associative. Assume that for every non-zero $a \in R$, the left multiplication map
$$ \lambda_a \colon R \to R \colon x \mapsto ax $$
is ...

**2**

votes

**0**answers

49 views

### Are there nilpotent Manin Triples?

Let $\mathfrak{g}$ be a Lie bialgebra and denote by $\mathfrak{d}$ the double of $\mathfrak{g}$, i.e. $\mathfrak{d}$ is a Manin triple. Are there known examples or conditions on $\mathfrak{g}$ for ...

**3**

votes

**1**answer

76 views

### Comodules of Cosemisimple Hopf Algebras

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...

**5**

votes

**0**answers

142 views

### Hemi-Semi Direct Product

In the category of racks (similarly quandles), instead of well-known semi direct product, we have hemi-semi direct product construction as seen on Wagemann & Crans.
As far as I know, semi direct ...

**-2**

votes

**1**answer

74 views

### Maximal commutative subrings of the endomorphism ring of a vector space

Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...

**2**

votes

**1**answer

143 views

### The center of a(n endomorphism) ring is a PID

Let $A$ be a torsion-free Abelian group, and let $\mbox{End}(A)$ be its endomorphism ring. Denote by $R\doteq\mathbf{Z}(\mbox{End}(A))$ the center of $\mbox{End}(A)$. What would be necessary and/or ...

**4**

votes

**0**answers

63 views

### An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true.
Is ...

**3**

votes

**1**answer

86 views

### An invariant submodule of a projective module

This is a basic question (not research level) which has already been asked on SE by someone else but doesn't yet have an answer so I'd like to repost it on MO.
Let $R$ be a commutative ring with ...

**7**

votes

**1**answer

157 views

### If a faithfully flat extension of dg/A_$\infty$-algebra is formal, is the original algebra formal (over positive characteristic)?

Proposition 6.2 of Formality of DG algebras (after Kaledin) by Lunts reads (with a few additions to clarify notation):
Let $k$ be a field of characteristic 0. Let $A$ be an $A_\infty$ algebra ...

**0**

votes

**1**answer

93 views

### Example of noncommutative central reduced rings which is not reduced

A ring $R$ is called central reduced if every nilpotent element is central. Ungor et al. math.RA 14 Dec 2013 has given an example of a commutative ring which is central reduced but not reduced. Can we ...

**0**

votes

**1**answer

74 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 ...

**1**

vote

**1**answer

81 views

### Jordan algebra of $3 \times 3$ quaternionic hermitian matrices

Let $\mathbb H = \mathbf H \otimes_{\mathbf R} \mathbf C$ be the tensor product of the quaternions with $\mathbf C$, and let $\mathcal J_3(\mathbb H)$ denote the set of $\mathbb H$-hermitian $3 \times ...

**5**

votes

**0**answers

157 views

### A question on symmetric functions

Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...

**-4**

votes

**1**answer

86 views

### Free algebra and ring of quotients [closed]

I am reading of example T.Y.Lam 'A first Course in Noncommutative Rings':
Let $R=\mathbb{Z}\langle x,y\rangle/(y^2,yx)$. To work with $R$, we shall confuse $x,y$ with their images in $R$. Thus,we ...

**1**

vote

**0**answers

63 views

### Does this system of equations have a closed form solution? [closed]

I am faced with the following system of equations and I'm looking for tools that allow me to characterize its solutions. The unknowns are $\{x_i\}$, $\{y_i\}$, and $\lambda$, for $i=1,...,N$. So there ...

**7**

votes

**0**answers

178 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 ...

**1**

vote

**0**answers

57 views

### Completion of an algebra

Based on arXiv:math/9802041v1, there is a definition for $NC$-filtration and $NC$-completion of an associated algebra over the complex numbers:
Let $R$ be an associative algebra and $R^{\rm Lie} = ...

**3**

votes

**0**answers

68 views

### False optima for control on Lie groups?

Consider the equation
$\frac{d Y_t}{dt} = (A + w(t)B) Y_t$
evolving on a compact semi-simple Lie group $G$ where $\frak{g}$ is the Lie algebra of $G$ and $A,B \in \frak{g}$ and:
$J:G \rightarrow ...

**1**

vote

**0**answers

87 views

### Transitivity for algebraic extensions of integral domains?

I'm trying to prove the following result. Let $R_1\subseteq R_2\subseteq R_3$ be integral domains such that $R_1\subseteq R_2$ and $R_2\subseteq R_3$ are algebraic extensions (note: I do not want to ...

**8**

votes

**0**answers

226 views

### Conjecture on matrix with reciprocal principal minors

Some notation: $A(\alpha|\beta)$ is the submatrix of $A \in \mathbb{R}^{n \times n}$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the ...

**8**

votes

**3**answers

482 views

### is every element in a C* algebra a product of normal elements?

I have the following question and since I am not an expert on C*-algebras, I thought I ask it here:
I know that in general the sum and product of normal elements need not be normal. It is even true ...

**2**

votes

**0**answers

65 views

### Is the positive part of an algebraic bilateral p-adic convergent power series algebraic?

Let $\mathbb{Z}_p \{ X\}$ and $\mathbb{Z}_p \{ X , X^{-1}\}$ be the henselizations of $\mathbb{Z}_p [X]$ and $\mathbb{Z}_p [ X , X^{-1}]$ with respect to the ideals $p\mathbb{Z}_p [X]$ and ...

**0**

votes

**1**answer

88 views

### Finitely generated projective modules over matrix rings [closed]

Is every (left) finitely generated projective modules over the matrix ring $M_n(\mathbb{C})$ isomorphic to a trivial module? Is there a good reference to look at this problem?
Apologies for asking ...

**2**

votes

**0**answers

164 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 ...

**7**

votes

**1**answer

182 views

### Hopfian modules

My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...

**2**

votes

**1**answer

71 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 ...

**1**

vote

**0**answers

56 views

### The projective resolution of a direct summand

For an $R$-module $M$ fix a projective resolution $P^\bullet\to M$. If $N$ is a direct summand of $M$, that is there is $L$ such that $M=N\oplus L$, then is there a projective resolution of $N$ which ...

**9**

votes

**0**answers

156 views

### Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module
$M = R / (ax + by + c) R$.
I am ...

**4**

votes

**0**answers

105 views

### Does a given bound quiver algebra admit an algebra analogous to the preprojective algebra of a path algebra?

Let $Q=(Q_0,Q_1,s,e)$ be a finite quiver.
In the formal construction of the preprojective algebra associated to $Q$, we set $\ Q^*_1=\{\alpha^*:y\rightarrow x| \alpha\in Q_1, \alpha:x\rightarrow y ...