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Questions tagged [noncommutative-algebra]

Non-commutative rings and algebras, non-associative algebras. Can be used in combination with ra.rings-and-algebras

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91 views

Dimension of hermitian rank at most $k$ matrices over quaternions

In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...
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Monoidal structure on left dg-modules over a brace algebra

Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
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41 views

A variation on Dixmier's counterexample concerning centralizers in $A_1$

This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some ...
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1answer
120 views

If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?

I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+...
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53 views

von Neumann regular ring homomorphisms

Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat. In particular, $\mathrm{id}...
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180 views

Constructing a noncommutative algebra from a commutative algebra

I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
3
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1answer
177 views

radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices

Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set $$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
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Relation between left projections

Let $A$ be a Baer *-ring. Let $x$ be in $A$, $L(x)$ is the left projection of $x$ that is the smallest projection with $L(x)x=x$. Q. Let $p,q$ are projections in $A$ with $p\leq q$. For a given ...
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35 views

The statue of a sequence of finite projections

Let $A$ be a Baer $*$-ring. Let $\{p_n\}$ be a sequence of finite projections in $A$. True or false? Suppose that there is no $N$ with $p_n=p_{n+1}$ for $n\geq N$. We have then $\inf_{1\leq n\leq ...
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something concerning finite projections

Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$). Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$. Q. Can we say that ...
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2answers
105 views

Upper triangular $2\times2$-matrices over a Baer *-ring

Let $A$ be a Baer $*$-ring. Let us denote $B$ by the space of all upper triangular matrices $\left(\begin{array}{cc} a_1& a_2 \\ 0 & a_4 \end{array}\right)$ where $a_i$'s are in $A$. Is $B$ ...
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What quantum groups admit quantum topography space structure?

Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...
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1answer
69 views

Strongly finite projections in $*$-rings

Let $A$ be a $*$-ring. Let us have some points: i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$. ii) On the set of projections, we write $p\leq q$ if $pq=p$. iii)...
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66 views

Transmission of finite projections

Let $A$ be a Baer*-ring. Let us denote $L(x)$ by the left projection of $x$ (the smallest projection with $L(x)x=x$). Let $p$ be a finite projection in $A$. Is $L(xp)$ a finite projection for every $...
6
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1answer
169 views

Is every (left) graded-Noetherian graded ring (left) Noetherian?

I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) ...
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138 views

Algebraic version of unilateral shift

It was confirmed that Wold-type decomposition can be extended from von Neumann algebras to Baer*-rings (see this paper). By algebraic tools the notion of unilateral shifts is successfully transmitted ...
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43 views

Free module over $H$-module algebra

Let $H$ be a finite dimensional Hopf algebra, $R$ be a $H$-module algebra and $V$ be a finite dimensional $H$-module such that $R\otimes_{k} V$ is a finitely generated $R$ module under the action: $r.(...
2
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1answer
105 views

Differential operators and rules Ore polynomial

(I have posed this question over at math.se but since there were no answers I hope it's okay to post here.) When dealing with (nonlinear) dynamical systems, one often deals with state space ...
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1answer
51 views

CAS implementing free algebras with involution

Is there any software that easily allows to make symbolic computations with involutions and homomorphisms? I need to define a product in an associative algebra with an (abstract) involution and ...
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82 views

Finitely generated submodules of projectives lie inside f. g. projectives?

Let $R$ be a (not necessarily commutative) ring. If $M$ is a finitely generated submodule of a projective module $P$, is there a finitely generated projective submodule $P'$ such that $M \subseteq P'...
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91 views

Hochschild homology and Chern character quiver with potential

I am a beginner in quiver theory so this question might not be suitable for mathoverflow. Let $(Q,W)$ be a quiver with potential and let $\Gamma$ be the Ginzburg DG-algebra associated to $(Q,W)$. Is ...
4
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1answer
111 views

number of indecomposable summands of an extension of two modules

I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of ...
2
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1answer
72 views

From socle of quotients to socle of ring itself

Let $I_1, \dots , I_n$ be ideals of a ring $R$ with identity having zero intersection. Assume that for some $x\in R$, $x+I_ i$ is an element of the right socle of $R/I_ i$, for each $ i=1,\dots , n$....
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40 views

Invertibility under base change for the Weyl algebra instead of for the polynomial algebra

From Lemma 1.1.8, we obtain the following: Assume that $R \subseteq S$ are commutative rings and $u: R[x,y] \to R[x,y]$ is an $R$-algebra endomorphism that has an invertible Jacobian, namely, $Jac(u(x)...
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65 views

When is $R/Soc(R)$ reduced?

Let $R$ be a ring with identity. It is readily checked that when the quotient $R/S_r$ is reduced, the nilpotent elements of $R$ fall into $S_r$, where $S_r$ is the right socle of $R$. Is the converse ...
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90 views

Language representation problem regarding non-commutative, non-associative algebras

Consider a sentence as a series of words with an associated set of labels that tell one how information is passed through the sentence - examples include combinatory categorical grammars or Lambek ...
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152 views

Hopf-Galois extensions where the “extension” is a module?

For $H$ a Hopf-algebra, an $H$-Hopf-Galois extension is a map of rings $\phi\colon\thinspace A\to B$ such that $H$ coacts on $B$ over $A$, $B\otimes_AB\cong B\otimes H$, and the cofixed points, or the ...
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Is $x \in A_1$ left algebraic over the subalgebra generated by $p$ and $q$, $[q,p]=1$?

Let $A_1:=A_1(x,y,k)$ be the first Weyl algebra over a field $k$ of characteristic zero, namely, the $k$-algebra generated by $x$ and $y$ with relation $yx-xy=1$. Let $f:(x,y) \mapsto (p,q)$ be a $k$-...
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m-systems and n-systems in topological rings

Note that throughout rings have a multiplicative identity and are not necessarily commutative Definition: Let $R$ be a ring and let $M\subseteq R$. Then, $M$ is an m-system iff for every $x,y\in ...
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2answers
178 views

A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero

Let $f,g \in \mathbb{C}[x,y]$. There is a well-known result, that can be found for example here, pages 19-20, that says the following: $f,g$ are algebraically dependent over $\mathbb{C}$ if and ...
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About an algebraic construction of a sheaf of formal microdifferential operators

While reading these notes by Victor Ginzburg on $D$-modules I found a certain construction of Microlocailzation in the algebraic setting which unfortunately doesn't seem to be elaborated on a lot in ...
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Intersections of generating sets of subalgebras

Let $A$ be a finitely generated, finitely presented, Noetherian, unital algebra over the complex numbers, which has no zero divisors. We do not assume that $A$ is commutative however. Moreover, let $...
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Which groups can occur as the group of units of finite-dimensional noncommutative algebras?

This is a continuation of a previous question: Connectedness of groups of units in finite-dimensional commutative algebras. Let $k$ be an algebraically closed field of characteristic $0$. Which ...
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1answer
202 views

Non-commutative regular sequences and non-commutative Koszul complex

I'm trying to understand the non-commutative Koszul complex, as can be found in Anick's nice paper "Non-Commutative Graded Algebras and Their Hilbert Series", J. of Algebra 78, (1982) and I'm stuck at ...
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group action on Tor groups of modules and smash product

I am trying to understand theorem 3.4.2 from the paper "Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over $\mathbf Z_{(p)}$ for representations with $p$-small weights" by ...
12
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1answer
416 views

Applications of cluster algebras

Why are so many algebraists nowadays interested in cluster algebras? (This is a rewording of one half of the closed question Cluster algebras and teichmuller theory.)
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1answer
102 views

If $\{f\in R[x]\:|\:f\text{ monic}\}$ is a right denominator set, is $\{f^i\:|\:i\geq 0\}$ a right denominator set also?

Let $R$ be a right (and left) Noetherian ring and $T=R[x]$ its polynomial ring. It was shown by Stafford that the set $S=\{f\in T\:|\:f\text{ monic}\}$ is a right denominator set. So my question is, ...
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192 views

Finitely generated skew-fields

There is a well known theorem saying that a commutative field that is finitely generated as a ring has to be finite (Kaplansky). Is the same true for non-commutative "fields" (usually called ...
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109 views

Is there a (nontrivial) known example of an algebra over a complete regular local ring with the following property?

I am working on some algebras over complete regular local algebras. But I am not sure whether such rings are worth to study. I am looking for some examples of these algebras. Let $(R,\mathfrak{m})$ be ...
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51 views

Reference request: Hecke agebra over non-commutative rings

I think the title sums it up quite well: Is it a useful idea to define the Iwahori-Hecke algebra over a non-commutative $k$-algebra? If so, what shape should the relations attain? Bonus question: ...
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175 views

Algebraic-closures of division rings

In what follows, $x$ is always taken to commute with the coefficient ring. This means that for any given polynomial, you can put the coefficients to the right or the left of $x$ as you please. This ...
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Do almost commutative flat degenerations induce equality in K-theory? (Or: Is the characteristic variety actually a support of a class in $K$-theory?)

I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not ...
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63 views

Is there an anti-commutator analog of Zassenhaus formula?

Is anyone familiar with an anti-commutator analog Zassenhaus formula? I have been able to find the anti-commutator analog of the BCH formula $$e^ABe^A= B + \{B,A\}+\frac{1}{2!}\{\{B,A\},A\}+ \frac{1}{...
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50 views

Any link between abelian $R/J(R)$ and 2-primal condition

Let $R$ be noncommutative unital ring such that each element of the quotient $R/Soc(R_R)$ is idempotent. If the nilpotent elements of $R$ form an ideal, is it true that the idempotents of $R/J(R)$ ...
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64 views

The going-up theorem for free extensions of almost commutative rings

I would like to know whether or not the going-up property holds for some classes of finite filtered extensions of non-commutative rings. Let $S \subseteq R$ be rings. The pair $(S,R)$ has the going-...
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1answer
100 views

If $H$ is essentially equimorphic to $K$, then is $H$ atomic only if so is $K$?

I will first state my question, and then give all the relevant definitions. Q. Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic only if so ...
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The group of automorphisms and anti-automorphisms of the first Weyl algebra

Let $k$ be a field of characteristic zero, and let $A_1=A_1(k)$ be the first Weyl algebra. It is well known (first proved by Dixmier, if I am not wrong) that the group of automorphisms of $A_1$, ...
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58 views

injective dimension of envelope algebras

Let $A$ be a connected graded algebra and $A^o$ the opposite algebra of $A$. Let $A^e=A\otimes A^o$. Suppose that $A$ has a finitely generated projective resolution as a graded $A^e$-module. My first ...
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1answer
67 views

Right localization of $R[x,x^{-1}]$ at monic $f\in R[x]$

Let $R$ be a right Noetherian ring and $S=\{f\in R[x]\;|\;f\text{ monic}\}$. It is a result of Stafford that $S$ is a right denominator set in $R[x]$, so in particular we can localize $R[x]$ at any $f\...
2
votes
1answer
79 views

Localising a right Noetherian ring at a set of regular elements

Let $R$ be a right Noetherian ring, and $S$ a multiplicative set consisting of regular elements where $1\in S$ and $0\not\in S$. Does the right ring of fractions $RS^{-1}$ exist? This is what I know ...