# Tagged Questions

**1**

vote

**1**answer

76 views

### Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?

**1**

vote

**0**answers

81 views

### Chinese Remainder Theorem

On a non-commutative ring we have this generalization of CRT.
I wonder if there is a statement involving only left or right ideals.
do you know any?

**2**

votes

**1**answer

135 views

### $Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$

I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say
...

**3**

votes

**2**answers

306 views

### noncommutative polynomials equality

Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$.
Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the ...

**4**

votes

**3**answers

406 views

### Units in a group algebra

Let k be a field and let G be a finite group. I would like to know if there is any nice description of the group of units in the group algebra kG. (If there is no nice answer in this generality, ...

**0**

votes

**1**answer

83 views

### $\mathrm{rk}_R M$ vs $\mathrm{rk}_S M$ - how nice need $R,S$ be?

Let $R\hookrightarrow S$ be Noetherian (noncommutative) rings without zero divisors with $\mathrm{rk}_{R} S < \infty$ (e.g. $S=R*G$ the crossed product of $R$ with a finite group $G$). Let $M$ be a ...

**4**

votes

**0**answers

117 views

### Injectivity criterion for surjective coalgebra maps: does it hold in full generality?

Let $k$ be a commutative ring.
Let $C$ be a filtered $k$-coalgebra. This means a $k$-coalgebra equipped with an increasing $k$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...

**5**

votes

**2**answers

189 views

### How big can a commutative subalgebra of Weyl algebra be?

Consider the smallest Weyl algebra $A_1=\{q,p; qp-pq=1\}$. It is known that there exist pairs of commuting elements, say $L$ and $M$, that obey various polynomial relations, e.g. elliptic curves. I ...

**11**

votes

**1**answer

318 views

### $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...

**14**

votes

**1**answer

355 views

### Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
What is the volume of the largest symmetric convex subset ...

**0**

votes

**1**answer

142 views

### Non-simple and non-unital rings with trivial centres

Let $R$ be an associative and non-unital ring. (Suppose that $R$ is $s$-unital, i.e. for each $x\in R$ there is $u,v\in R$ such that $ux=xv=x$.)
It is not difficult to show that if $R$ is a simple ...

**9**

votes

**0**answers

221 views

### Does there exist a Noetherian ring of finite injective dimension but higher Krull dimension?

Definition: a (not necessarily commutative) left and right Noetherian ring $R$ is said to be Auslander-Gorenstein if
(i) $R$ has finite left and right injective dimension (in which case it turns out ...

**2**

votes

**1**answer

139 views

### Algorithmically finite-dimensional (noncommutative) algebras.

Can anyone help to find some information about these structures?

**9**

votes

**0**answers

441 views

### Is “being a full ring of quotients” a Morita invariant property?

Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...

**1**

vote

**0**answers

194 views

### Annihilator ideals

For an ideal $I$ of a ring $R$ with identity, let $r(I)=\{r\in R: Ir=0\}$ and $l(I)=\{r\in R: rI=0\}$.
Question: If for any two ideals (two-sided ideal) $I, J$ of $R$, we have $l(I)+l(J)=l(I\cap J)$, ...

**8**

votes

**1**answer

160 views

### Are annihilation modules in the quantum torus necessarily principal?

I hope that my question yields some standard fact from (noncommutative) ring theory. In discussions with other graduate students, we have outlined some approaches to tackling the question, but ...

**1**

vote

**0**answers

72 views

### Finite dimensional consistently graded Lie superalgebras of depth greater than 2

Victor Kac, in the paper
"Classification of infinite-dimensional simple linearly compact Lie superalgebras", http://www.mat.univie.ac.at/~esiprpr/esi605.pdf
writes at the beginning of section 5 ...

**5**

votes

**0**answers

402 views

### A reference on semisimple linear algebra

Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...

**5**

votes

**2**answers

943 views

### Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. I'm seeking a kind of Schur's lemma, with $\mathrm{Hom}_R (M,N)$ replaced by $M \otimes_R N$. So my questions are:
Can ...

**1**

vote

**1**answer

100 views

### Projective dimension over hypersurface

Let $R$ be (not necessarily commutative) ring and $S$ a simple right $R$-module. Let $f\in Ann(S)$ be normalizng and a non-zero divisor. Is it always true that
$$
pdim_{R}(S)=pdim_{R/(f)}(S)+1?
$$

**2**

votes

**1**answer

188 views

### Global dimension of quantum $\mathbb{P}^{n}$

Let $k$ be a field. Given a (not necessarily commutative) $k$ graded ring $A$, M. Artin and J.J. Zhang introduced a notion of "noncommutative projective scheme" $Proj(A)$ in this paper. It is defined ...

**7**

votes

**2**answers

960 views

### Global dimensions of non-commutative rings

This is related to my previous question: When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau? I now would like to know the global dimension of the ring $R=\mathbb{C}\langle ...

**13**

votes

**4**answers

856 views

### What's the name for the analogue of divided power algebras for x^i/i?

I recently came across divided power algebras here: http://amathew.wordpress.com/2012/05/27/lazards-theorem-ii/ It interests me because the free divided power algebra on one variable $x$, where ...

**0**

votes

**1**answer

221 views

### A Version of Nullstellensatz for Rings of DÄ°fferential Operators

Here is one of the classical versions of the nullstellensatz: Let $K$ be a field and let $\mathfrak{m}$ be a maximal ideal of the polynomial ring $K[T_1,\ldots,T_n]$. Then ...

**4**

votes

**4**answers

532 views

### Noncommutative Localization of a Ring : Complete Construction

I've been looking for the following construction in the literature, but I've only been able to find (very) partial proofs or proofs of special cases.
Let $R$ be a non-commutative ring and $S$ a ...

**3**

votes

**0**answers

105 views

### 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 ...

**0**

votes

**1**answer

108 views

### 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?

**12**

votes

**1**answer

395 views

### Is the ring of quaternionic polynomials factorial?

Denote by $\mathbb{H}[x_1,\dots,x_n]$ the ring of polynomials in $n$ variables with quaternionic coefficients, where the variables commute with each other and with the coefficients. Two polynomials ...

**4**

votes

**1**answer

208 views

### Prime ideals in maximal orders (1- and 2-sided)

I am an arithmetic geometry graduate student, and I find myself needing to learn about factorisation in orders in division algebras. I know something aout algebraic number theory and commutative ...

**2**

votes

**0**answers

310 views

### Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task:
I have two graded (generally ...

**2**

votes

**2**answers

277 views

### Are morphisms of finite length modules determined by the behaviour of the simple modules?

Assume we have a noncommuative ring $R$ with exactly 2 non-isomorphic simple left modules $S_1$ and $S_2$ (up to isomorphism) and an $R$-bimodule $M$, which switches the simples, i.e. $M\otimes_R ...

**5**

votes

**1**answer

1k views

### 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 ...

**33**

votes

**1**answer

784 views

### 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 ...

**31**

votes

**1**answer

2k views

### 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 ...

**1**

vote

**1**answer

606 views

### Local Rings problem

$\newcommand{\End}{\operatorname{End}}$
let $R$ be a local ring, $\varphi\in \End(R_{R}^{2})$,
$\overline{\varphi}\in \End(\overline{R}_{\overline{R}}^{2})$,
$\overline{R} =R/J(R)$ , $J(R)$= ...

**3**

votes

**2**answers

371 views

### 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 ...

**5**

votes

**2**answers

463 views

### 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 ...

**15**

votes

**8**answers

3k views

### 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 ...

**4**

votes

**1**answer

1k views

### Are there any finitely generated artinian modules that are not notherian?

It is well known that for rings, Artinian implies Noetherian (the famous HopkinsāLevitzki theorem) and it is also well known that there are Artinian modules which are not Noetherian. A simple example ...

**1**

vote

**0**answers

311 views

### The sum of a nilpotent left ideal and a nil left ideal

In class, we recently saw that the sum of 2 two-sided nil ideals is a nil ideal. We were asked to show that the sum of a niplotent left ideal and a nil left ideal is a nil left ideal.
I am having ...

**29**

votes

**7**answers

3k views

### 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 ...

**6**

votes

**2**answers

1k views

### 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 ...

**2**

votes

**2**answers

368 views

### on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring

Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some ...

**9**

votes

**2**answers

959 views

### 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 ...

**2**

votes

**0**answers

147 views

### 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 $sr'=r$, ...

**21**

votes

**1**answer

640 views

### Idempotents in Rings of Differential Operators

Differential Operators on General Commutative Rings
Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator ...

**7**

votes

**1**answer

457 views

### Mathematical software for computing in integral group rings of discrete groups?

I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where ...

**3**

votes

**0**answers

143 views

### Global dimensions of orders over non-Gorenstein centers

This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that ...

**3**

votes

**1**answer

611 views

### Semiprime (but not prime) ring whose center is a domain

The center of a prime ring is a domain and the center of a semiprime ring is reduced.
Now I have no evidence to believe that if the center of a semiprime ring R is a domain,
then R has to be a ...

**10**

votes

**2**answers

1k views

### Maximal ideal that annihilates entire ring

Does there exist a ring R with a nonzero maximal ideal M such that R^2=R and MR = RM = 0?
Here R is associative but does not have an identity (obviously). It seems a simple enough question but I'm ...