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

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7
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153 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
5
votes
0answers
43 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
5
votes
0answers
78 views

NCG with all noncommutativity in a nilpotent ideal

While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative ...
5
votes
0answers
170 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
103 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 ...
5
votes
0answers
180 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
173 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
215 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. ...
4
votes
0answers
382 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
70 views
+50

What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $S \leftarrow ...
3
votes
0answers
61 views
+50

A Problem for Nil-Ideals

Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for non-commutative rings I think this may not be true. ...
3
votes
0answers
52 views

What properties do the rings of infinite, upper-triangular matrices have?

I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field. But ...
3
votes
0answers
53 views

Prove that under some condition the commutator subgroup contains not only commutators

I'm trying to prove the following statement: Let's assume $ G $ is a finite group. Let $ Z(G) $ denote its center, $(G : Z(G))$ the index of $ Z(G) $ in $ G $ and $ [G; G] $ the commutator subgroup ...
3
votes
0answers
163 views

Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
3
votes
0answers
45 views

$\operatorname{Br}(\Bbb Q_{47})$

I'm looking for division algebras over $\Bbb Q_{47}$. I guess my best bet is to calculate the Brauer group $\operatorname{Br}(\Bbb Q_{47})$. What's the best way of performing this calculation? Should ...
3
votes
0answers
21 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
109 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
99 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
129 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
33 views

Bochner-style theorem for SO(3)

Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar ...
2
votes
0answers
40 views

Homological Conjectures

Let The strong Nakayama conjecture : If $M \in \rm{{mod\mbox{-}}}R$ and $\rm{Ext}^i(M,R)=0$ for $i \geq 0$, then $M$ is zero. The generalized Nakayama conjecture If $S$ is a simple module and ...
2
votes
0answers
23 views

Simultaneous triangular form for nil algebras over division ring

Let $D$ be a division ring. Let $V$ be a finite dimensional module over $D$, let $I \subseteq\operatorname{End}_D(V)$ be a $D$-submodule on both sides (I mean a subgroup closed by both on left and ...
2
votes
0answers
33 views

How is this commutator property $[a,b]=bx$ called?

If two elements $a,b$ commute like this $[a,b]=bx$ for some $x$ so that it can rewriten as $$ ba=a(b-x), $$ is there a term for such property? It is as if $a$ and $b$ almost commuted (but not quite) ...
2
votes
0answers
30 views

Tensor product and opposite rings

Suppose $A$ is a ring, $M$ is a right $A$-module, and $N$ is a left $A$-module. In this situation, we can form the tensor product, $M\otimes_A N$, and this is an abelian group (and even a ...
2
votes
0answers
47 views

Endomorphism rings of a $k$-algebra

Let $k$ be a field and $R$ be the $3$-dimensional $k$-algebra $\left [\begin{array}\ k & k \\ 0 & k \end{array} \right ]$. I have two conjectures: (1) Is any simple right $R$-module has ...
2
votes
0answers
40 views

Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
2
votes
0answers
40 views

Ordered rings $R$ with $ab = ba$, $ab \leq ba$ or $ba \leq ab$ for all $a, b \in R$?

Ordered rings $R$ with $ab = ba$, $ab \leq ba$ or $ba \leq ab$ for all $a, b \in R$? Are there any good examples that are not also commutative rings? I can't seem to think of any.
2
votes
0answers
76 views

Simple Modules over the Weyl Algebra

Let $k$ be a field of characteristic zero and let $A_1=k\langle x,y| \, xy-yx=1 \rangle$ be the Weyl algebra. Is there a (more or less explicit) possibility of writing down all simple modules over ...
2
votes
0answers
54 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 ...
2
votes
0answers
81 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
41 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
39 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
78 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
95 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
94 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
54 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
90 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
52 views

What do we call the ring homomorphism $R \rightarrow \mathrm{End}_{\mathbf{Ab}}(X)$ associated with an $R$-module $X$?

First, a convention: given an abelian group $X$, write $\mathrm{End}_{\mathbf{Ab}}(X)$ for the set of all group homomorphisms $$X \rightarrow X.$$ Now let $R$ denote a ring. Question. Given an ...
1
vote
0answers
26 views

If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.

Let $I$ be a left ideal in a semiprime ring $R$. If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.
1
vote
0answers
20 views

On algebraicity of a formal power series

In his paper "Noncommutative identities" M. Kontsevich states the following: Theorem 2. For any $P=P(x,y)=1+\cdots\in\mathbb{C}[x,y]$ expand ...
1
vote
0answers
26 views

If $R$ is a noncommuative Noetherian ring, is it true that $R[[x]]$ is a Noetherian ring?

I am looking for a reference in order to answer the following question: If $R$ is a noncommutative Noetherian ring, is it true that $R[[x]]$ is a Noetherian ring? The answer is well-known in the ...
1
vote
0answers
54 views

If $R$ is generated by idempotents, then $\text{Ann}(R)=0$?

Let $R$ be a ring (not necessarily commutative or unital) which is generated by idempotents. I'd like to know if $\text{Ann}(R)=0$ must holds. Here I use $\text{Ann}(R)$ to denote the set of all ...
1
vote
0answers
12 views

Construction of the decomposition of semisimple algebra

I'm a theoretical phycysist and I'm working on the theory of quantum information, in which some problems are connected with properties of matrix algebras. My question is: If we know, that given ...
1
vote
0answers
37 views

Projectivity of a certain module

Let $M$ be a generator for the category of left $R$-modules, and let we have an $R$-epimorphism $h$ from $R^{(X)}$ to an $R$-module $P$ which is projective relative to $M^{(X)}$ ($X$ is a set). I want ...
1
vote
0answers
51 views

Inverse of a $2\times2$ matrix with noncommuting entries

I want to invert a $2 \times 2$ matrix with quaternionic entries. Since non-commutativity, the determinant is not well defined and I've seen in http://en.wikipedia.org/wiki/Quaternionic_matrix that, ...
1
vote
0answers
66 views

Commuting exponentials of non-commuting matrices

For two non-commuting matrices $A,B \in M(2,\mathbb{K})$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, can be shown that: $$ e^C=e^{A+B}=e^Ae^B=e^Be^A \iff \begin{cases} ...
1
vote
0answers
61 views

Total ordering on the free group

The free groups can be totally (bi-)ordered. This paper shows how to do it (page 4). In short, you embed the group in multiplicative structure of the ring of power series in non-commuting variables, ...
1
vote
0answers
33 views

Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
1
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0answers
30 views

Considerations for moving a function inside or outside of an integral

Excluding the possibility that $A(t)$ is the limit of a sequence, are there any special considerations I should be concerned with regarding the following assertion: Let $A(t)$ be an $n\times n$ ...
1
vote
0answers
38 views

Example of a regular element with a commutative quotient

Whats an example of ring $A$ which is not commutative and contains an element $x\in Z(A)$ such that the left multiplication by $x$ is an injection and $x$ is not a unit and $A/(x)$ is commutative?