For questions about rings which are not necessarily commutative and modules over such rings.

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6
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0answers
62 views

Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
6
votes
0answers
137 views

A few questions about a specific ring

My question is kinda long, so please bear with me... And I only need hints to get me started. So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & ...
5
votes
0answers
151 views

When is $\mathbb{Z}\Gamma$ a left Noetherian ring?

Denote $\Gamma$ to be a countable discrete group, let $\mathbb{Z}\Gamma$ to be its integer group ring. A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals.(c.f. ...
5
votes
0answers
156 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
5
votes
0answers
151 views

Problem 24 of section 2 of *Noncommutative Algebra* by Farb & Dennis

Problem 24 of section 2 of Noncommutative Algebra by Farb & Dennis states: Let $R$ be an artinian ring and let $G$ be a finite group. Show that $R[G]$ is semisimple if and only if $R$ is ...
5
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0answers
172 views

Examples of Noncommutative Noetherian Rings in which Lasker-Noether Fails?

I'm writing a paper on Emmy Noether for my introductory Abstract Algebra class, and I'm looking for examples of noncommutative Noetherian rings in which the Lasker-Noether theorem fails to hold. ...
3
votes
0answers
20 views

Reference for proof of homotopy invarance of Cyclic cohomolgy

I'm looking for a good reference for a proof of the homotopy invariance of cyclic (co)homology. I'm following a refernce book by Joachim Cuntz, the proofs are ommited therein, or only shown in the ...
3
votes
0answers
42 views

Is there an analog of Cech complex for local cohomology over noncommutative rings?

Let $A$ be a noetherian ring, and let $I\subseteq A$ be an ideal. Suppose $I$ is generated by $a_1,\dots,a_n$. Let $M$ be a left $A$-module. If $A$ is commutative, then one can compute the derived ...
3
votes
0answers
188 views

On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
3
votes
0answers
89 views

Applications of Govorov-Lazard Theorem?

The Govorov-Lazard Theorem states that a (right) module over an unital ring is flat iff it is a direct limit of finitely generated free (right) modules. I wonder if this theorem has interesting ...
3
votes
0answers
84 views

Computation of determinant of a matrix with elements from an arbitrary commutative ring

The cofactor formula for computing the determinant of a matrix is applicable when elements of the matrix are from a commutative ring. However, the complexity of this method is extremely high and I ...
3
votes
0answers
122 views

Integral forms of loop algebras.

The question following is about integral forms for semisimple Lie algebras and loop algebras constructed from them. Let $\frak g$ a finite-dimensional Lie algebra over $\mathbb C$ and $L(\frak ...
2
votes
0answers
18 views

Smooth algebras and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
2
votes
0answers
40 views

Does quotient commute with localization for non-commutative rings?

Following on from the question Does quotient commute with localization?, I'm interested in doing the same sort of thing but over non-commutative rings. Is there a non-commutative analogue of the ...
2
votes
0answers
33 views

Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
2
votes
0answers
33 views

Regarding splitting in the Brauer group over fields of prime characteristic.

Let $k$ be a field of characteristic $p$, and let $K=k^{1/p}$. We want to show : $[A]\in\rm Br(k)$ splits over $K$, i.e. $[A\otimes_k K]=1\in\rm Br(K)$ iff $p. [A]=1\in\rm Br(k)$. Here $[A]$ denotes ...
2
votes
0answers
54 views

on rings with an inner product

For this question, "ring" includes having a $1$ and "subring" includes having the same $1$. Let $F$ be a ring, let $E$ be an ordered subring of $F$ with no non-trivial zero divisors, let $\: ...
2
votes
0answers
82 views

Inverse function of product of exponential matrices

I am looking for the value of $\mathbf{X}$ in a function of the type \begin{align} (\mathbf{X}-\mathbf{A})e^{\mathbf{X}}e^{-\mathbf{A}} = \mathbf{B} \end{align} where ...
2
votes
0answers
68 views

Defining the Rank of a Projective Module

I am trying to understand the definition of rank for a projective module over a noncommutative ring. The definition I am using is: A sufficient condition for the rank of a free module over a ring ...
2
votes
0answers
48 views

Left ideals of central simple algebra generated by symmetric element

Let $F$ be a field of characteristic $\neq 2$. Suppose $(A,\sigma)$ is central simple algebra over a field $F$ with an orthogonal involution. Assume $(A,\sigma)$ is non-split. How to show that every ...
2
votes
0answers
79 views

Non-commutative integral extensions?

In Commutative algebra there is a notion of an integral extension: Let $P$ be a subring of $R$. Then $R$ is the integral extension of $P$ if each element of $R$ is a root of a monic polynomial with ...
1
vote
0answers
30 views

Tensor product of two simple modules

Let $k$ be a field of arbitrary characteristic. Suppose that $A$ and $B$ are finite dimensional $k$-algebras. Let $S$ be a finite dimensional simple $A$-module and let $T$ be a finite dimensional ...
1
vote
0answers
26 views

Tensoring and retaining projectiveness

Let $A$be a unital associative ring, If $A$ is not a projective $A$-bimodule, however $A\otimes A$ is a projective $A$-bimodule, can we conclude that $A\otimes A \otimes A$ is also projective?
1
vote
0answers
26 views

Coproducts and Hochschild

I $\{X_i\}$ is a small family of associative $\mathbb{C}$-algebras and $X$ is their free product. Then I have two questions: 1) Why is $X$ their coproduct? 2) Is the Hochschild homology of X ...
1
vote
0answers
25 views

Grading and commutators

If $R$ is a unital associtaitve commutative ring, then can any $R$-algebra $A$ may be filtered as $A_0:=A$ and $A_{i+1}:=(A_i,A_i)$ where $(-,-)$ is the commutator of $A$ with respect to $A$'s ...
1
vote
0answers
83 views

Relation between finite stable rank and IBN (invariant basis number)

For R is any ring has an identity, we known if R has stable rank one, then R is "weakly finite" (or "stably finite," all matrix rings over R are Dedekind finite) and this implies R has IBN . But ...
1
vote
0answers
44 views

Integral elements with predescribed properties in quaternion orders

In the course of doing some calculations I have found myself wanting to answer the following question: Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...
1
vote
0answers
26 views

Skew polynomial algebra and deformation

Let $R$ be an associative unital $k$-algebra. If $\alpha \in End_k(R)$ and $\delta$ is a $\alpha$-derivation, then one can define the skew polynomial algebra $R[x; \alpha,\delta]$ by letting $ax = x ...
1
vote
0answers
41 views

Enveloping Algebra equal to algebra

Let $R$ be a unital associative ring, $A$ be an associative $R$-algebra of finite dimension, and $A^e$ its enveloping algebra. What are the requirements on $A$, so that $A^e \cong A$ (as ...
1
vote
0answers
27 views

Every onesided nilideal of a right noetherian ring is nilpotent.

Suppose $R$ is a right noetherian ring. Prove that every onesided nilideal is nilpotent. I try to use this theorem: If R is a commutative Ring and I is nilideal of R and also I is finitely ...
1
vote
0answers
35 views

For certain prime ideal $\wp$, $(\wp^e)^c=\wp$?

Let $R\subset S$ be noncommutative rings, and let $\wp$ be a prime ideal in $R$, denote $\wp^e$ to be the extension ideal of $\wp$ in $S$, then my question is: Does $(\wp^e)^c:=\wp^e\cap R=\wp$ hold ...
1
vote
0answers
110 views

Existance of inverse operators for Hermitian adjoint operators

I have one assumption that I can't prove. Maybe there are some other conditions which I didn't take into account. My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and ...
1
vote
0answers
87 views

Left-modules over a bialgebra form a monoidal category

Let $B = (B, \nabla, \eta, \Delta, \epsilon )$ be a bialgebra over a commutative ring $k$. Let $M$ and $N$ be two left $B$-modules. Then the tensor product $M \otimes_k N$ becomes a left $B$-module ...
1
vote
0answers
143 views

All finite-dimensional simple modules are $1$-dimensional

Let $A$ be a (non-commutative) $k$-algebra, where $k$ is an algebraically closed, characteristic zero field. Let $M$ be a finite-dimensional simple $A$-module. If $A/\operatorname{ann}(M)$ is ...
1
vote
0answers
105 views

Unit group of quotient of noncommutative polynomial ring

In this recent post the original question led people to look for rigid, noncommutative rings. (Rigid means that the only endomorphisms are zero and the identity). Several (somewhat complicated) ...
1
vote
0answers
50 views

Nil ring version of Burnside's problem

The famous Burnside group of exponent $n$ over a set $G$ of generators is the group defined by generators $g\in G$ and relations $h^n=1$ where $h$ is any product of elements of $G$. Likewise, let ...
1
vote
0answers
106 views

PBW Theorem applied to graded Lie algebras

Fix a $\mathbb Z_+^n$-graded Lie algebra ${\frak a}=\oplus_{r \in\mathbb Z_+^n}^{} {\frak a}[r]$ such that ${\frak g}:={\frak a}[0]$ is a finite-dimensional semisimple Lie algebra over the complex ...
0
votes
0answers
17 views

Infinite series of nested commutators

I'm trying to show the following: If $S_i$ are a set of three matrices such that $$ [S_i, S_j] = \epsilon_{ijk} S_k $$ then $$\exp\big( \alpha_i [S_i, \cdot]\big) S_j = (\exp (M) \vec{S})_j$$ ...
0
votes
0answers
28 views

pullback along a homomorphism

I am studying the grand book of T.Y. Lam, Lectures on Modules and Rings, in which the expression of pullback of some superset $B$ of a set $A$ along a homomorphism $\phi$ from some set $T$ to $A$ is ...
0
votes
0answers
6 views

DCC on prime ideals in noncommutative noetherian ring

I know that commutative Noetherian rings satisfy the DCC on prime ideals, and that this is not true in general for NC rings which are Noetherian on only one side. Are there any intermediate results? ...
0
votes
0answers
15 views

Commutativity of comodules

If C is a cocommutative R-coalgebra, R is some commutative semi-simple artinian ring and A and B are C-bicomodules, then is $A\otimes_R B \cong B \otimes_R A$ as $R-modules$. However, this also true ...
0
votes
0answers
79 views

Rings, annihilators and (maximal) ideals

Let $R$ be a unital, associative, non-commutative ring. If $P$ is an ideal of $R$, what is the annihilator of quotients $R/PR$ and of $R/P$? Does something change if $P$ is supposed to be a maximal ...
0
votes
0answers
45 views

Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left) noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by finite set $x_1,...,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. Hence a Poisson ...
0
votes
0answers
15 views

Gaussian for Grassmann variables

Let $(\theta,A\theta)=\theta_i A_{ij}\theta_j$ where $A$ is some $(2\times2)$ antisymmetric matrix. I want to generalize the following $$I(A) =\int d\theta_1d\theta_2~ ...
0
votes
0answers
26 views

Basic facts about finitely-generated noncommutative algebras

I don't really have any feeling for non-commutative rings, so I just wanted to check that I'm sane. A simple "yes" or "no" will probably do, unless I'm doing something incredibly stupid in which case ...
0
votes
0answers
45 views

Non-constant solution of $f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$ in rings

The answers to this question prove that over an integral domain the functional equation $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ has only solutions where one of $f$ or $g$ is constant. The proofs make ...
0
votes
0answers
59 views

Show that Weyl algebra is noetherian

Let $k$ be a field. I want to show that the ring $D=k\left[x_1,x_2,\dots,x_n,\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\dots,\frac{\partial}{\partial x_n}\right]$ which acts on ...
0
votes
0answers
21 views

A question about C* probability spaces.

If I have $C(\Omega)$: the commutative C*-Algebra of all continuous random variables then the C*Algebra $C(\Omega)$ acts on the Hilbert space $L^2(\Omega)$ by left multiplication, and so we have the ...
0
votes
0answers
38 views

Graded comodule isomorphism

Let $R$ be a semisimple ring. Let $C$ be a graded, connected $R$-coring (i.e. coalgebra in the monoidal category of $R$-bimodules), $C=\oplus_{n\geq 0} C_n$, such that $C_0 \simeq R$. Denote by $C_+$ ...
0
votes
0answers
21 views

Abstract Algebra Question concerning $o(3)$ generators in a Hopf Algebra

Define $q = e^{\eta}$, such that $q$ is an arbitrary parameter. I need to show the following; $$0 = -q^{\mp 1} L_{\pm} q^{L_0} + q^{L_0} L_{\pm}$$ Where $L_{\pm}, L_0$ are the $o(3)$ generators. ...