# Tagged Questions

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### Making an algebra the uniqe maximal one-sided ideal in another unital algebra

If $R$ is an algebra without a unit, then the standard unitisation $R^\sharp$ can have maximal one-sided ideals other than $R$. Thus, it is natural to ask about the following. Let $R$ be an algebra ...
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### The octonion equations

A good treatment have been given to the quaternion equations. Indeed, Ivan Niven in his paper Equations in Quaternion given in this link http://jones.math.unibas.ch/~massierer/algebra-hs11/niven(...
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### Center of universal enveloping algebra of nilpotent lie algebra

Let g be a finite dimensional nilpotent lie algebra over a field k of characteristic zero. Let U(g) be the universal enveloping algebra and Z(g) be its center. Denote by Z_1(g) the augmentation ideal ...
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### Local algebras with small maximal left ideals

Is there an infinite-dimensional, non-commutative complex local algebra $A$ (which is not a field) with the (unique) maximal left-ideal finitely generated as a left ideal? Or as a right ideal?
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### Commutator Baker-Campbell-Hausdorff formula

Consider the Baker-Campbell-Hausdorff formula $\Phi(X,Y)\in\mathbb{Q}\langle\!\langle X,Y\rangle\!\rangle$ in non-commutative variables. Define $X*Y:=\Phi(X,Y)$ and $[X,Y]=(-X)*(-Y)*X*Y$, and then (as ...
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### Algebra - Decomposition of a matrix polynomial

Dear All, This is related with a problem that I'm trying to solve on my PhD dissertation in econometrics, and I thought that some mathmatician can know the answer. What is known about a possible ...
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### Invertible matrices over noncommutative rings

Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples? The question popped up ...
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### What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
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### Binomial Expansion for non-commutative setting

What could be a reference about binomial expansions for non-commutative elements? Specifically, where can I find a closed formula for the expansion of $(A+B)^n$ where $[A,B]=C$ and $[C,A]=[C,B]=0$? ...
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### Expressing a element of a Matrix subgroup in terms of subgroup generators

I'm no (computational) algebraist, and my searches have been pretty unyielding (probably due to the vast amounts written on the key words), but perhaps someone may know if this is possible, and if so, ...
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### G = [G,G] with two generators

Is it true that groups $\langle a,b \mid a^n b^k=b^ka^{n+1}, b^la^s=a^sb^{l+1}\rangle$ are non-trivial for almost all (in any sense:))) $n,k,l,s\in\mathbb N$?
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### The Jacobson radical of an infinite dimensional algebra

Does any one know the Jacobson radical of the path algebra of the following quiver? $$\bullet \leftrightarrows \bullet$$ How many simplerepresentations of it are there? Is there any software that ...
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### Gröbner/SAGBI bases for non-commutative setting

It is well known that SAGBI/Gröbner bases are important for commutative and non-commutative algebra. The references for commutative scenery is ample and vast, but I am in trouble to find a good ...
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### Central division and quaternion algebras

I would like to know if there are some central-simple algebras $D_1$, $D_2$ and $D_3$ over a field $k$ satisfying the following properties : $ind(D_1)=exp(D_1)=4$ ($ind$ is the Schur index and $exp$ ...
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### A Question on Koszul duality and $B(\infty)$ structures on $HH^*$

The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type. There is an equivalence of Gerstenhaber ...
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### Local Rings problem

$\newcommand{\End}{\operatorname{End}}$ let $R$ be a local ring, $\varphi\in \End(R_{R}^{2})$, $\overline{\varphi}\in \End(\overline{R}_{\overline{R}}^{2})$, $\overline{R} =R/J(R)$ , $J(R)$= ...
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### An example where finitistic dimension does not equal right global dimension?

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...
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### Simple Ore extensions

Let $R[x;\sigma,\delta]$ be an Ore extension, where $R$ is an associative and unital ring and $\sigma : R\to R$ is a (not necessarily injective!) ring endomorphism. (In the literature it is often ...
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### A potential resolution of $R/r$

The DGA For $k$ some field, let $R$ be a $k$-algebra, and let $r\in R$. Define a differential graded algebra $\mathbf{R}_r$ as follows. As a graded algebra, it is isomorphic to $R\langle t\rangle$, ...
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### Tensor power- Notation question

Hi everyone I have a notational question, which is written usually in papers, but I can not figure it out what could be. Let $M$ be an $A$-module. I have seen this notation $$M^{\otimes -n}$$ I ...
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### Simplest examples of rings that are not isomorphic to their opposites

What are the simplest examples of rings that are not isomorphic to their opposite rings? Is there a science to constructing them? The only simple example known to me: In Jacobson's Basic Algebra (...
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### What makes a theorem *a* “nullstellensatz.”

I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...
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### Jacobson radical = intersection of all maximal two-sided ideals

I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can ...
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### Unit ideal in non-commutative rings [closed]

In a non-commutative ring (with identity), is it possible for an element which does not possess left or right inverses to generate the entire ring? i.e. $(r)=R$, where (r) is the two-sided ideal ...
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### Finite Homological Dimension of R/P for all P for module finite non-commutative rings

I have a reasonably precise question which I hope is clear enough to get a nice answer. Let R be a Noetherian non-commutative ring which is finite as a module (and flat/free if it helps) over it's ...
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### Why does the Grothendieck group $K_0(R)$ of a ring not depend on our choice of using left modules instead of right modules?

I am under the impression that in the definition of the Grothendieck group $K_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K_0$ construction to the exact category of ...
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### Compute Lie algebra cohomology

Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation? What if the Lie algebra is finite dimensional? In my case I would like to be able to ...
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### How to work with co-multiplication?

Let $C$ be a coalgebra and $\Delta: C \to C\otimes C$ a co-multiplication map. Then, due the co-associative property we can consider $\Delta^m$. But how is defined $\Delta^{m}: C \to C^{\otimes m}$??? ...
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### Identifying a sequence of polynomials

Studying a specific quantum cluster algebra, I have come across the following sequence of polynomials : $$X_1$$ $$q^{-1/2}(X_1X_2-1)$$ $$q^{-1/2}(X_1X_2X_3-X_3-X_1)$$ q^{-1}(X_1X_2X_3X_4-X_3X_4-...
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### Does there exist any massive proper $C^*$-subalgebra?

Definition 1: Suppose $B$ is a $C^*$-algebra. $A$ is massive $C^*$-subalgebra of $B$ iff 1. $A$ is a subalgebra of $B$; 2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...
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### Pulling out factors in a Noetherian Domain

Let $R$ be a Noetherian domain (not-necessarily commutative), and let $S$ be a Noetherian subring of $R$. An element $r\in R$ is left $S$-irreducible if, for any $s\in S$ and $r' \in R$ with $sr'=r$, ...
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### Set of invertible operators in B(H) is connected. Is it true? Is there a reference?

Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
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### Morphisms of a simple sheaf over an algebra to its double dual

Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O_S$-module of finite rank. Let $M$ be a coherent and ...
Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$. Under what ...