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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1answer
132 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 $SL_n$ ...
-4
votes
0answers
78 views

Everywhere holomorphic functions over C [on hold]

Let $f, g$ be two everywhere holomorphic functions over ${\Bbb C}$. We consider the local representation of $f, g$ at the origin of ${\Bbb C}$, i.e. $z = 0$. That is, we can consider $f, g \in {\Bbb C}...
2
votes
1answer
113 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
1answer
248 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
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0answers
85 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 ...
-1
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0answers
61 views

When will the spectral radius of a matrix reach its minimum?

Let $A$, $B$ be two $n\times n$ matrices. How to determine $s,t\in\mathbb{N}^+$ such that the spectral radius of $A^sB^t$ will reach its minimum?
4
votes
1answer
88 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
0answers
93 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
1answer
242 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
1answer
211 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
1answer
85 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
1answer
178 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
0answers
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
1answer
122 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
0answers
123 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
0answers
163 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
0answers
69 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) \...
11
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0answers
166 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
0answers
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
0answers
32 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
1answer
112 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
1answer
71 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
2answers
230 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
0answers
37 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
2answers
494 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 $C(X)$...
8
votes
1answer
120 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
1answer
230 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
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0answers
52 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
0answers
65 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
1answer
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
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0answers
145 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
1answer
526 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
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0answers
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
0answers
258 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
0answers
147 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
1answer
179 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
1answer
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
2answers
194 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
2answers
92 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
0answers
51 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
1answer
79 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
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0answers
144 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
1answer
75 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
1answer
146 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
0answers
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
1answer
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 ...
8
votes
1answer
163 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
1answer
95 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 ...