Questions about rings that are not necessarily commutative.

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4
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0answers
151 views

Generation of cohomology of graded algebras

Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...
6
votes
2answers
151 views

Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial?

Let $G$ be a (possibly infinite) group. Let $\mathbb{Z}[G]$ be its integral group ring and let $P$ be a finitely generated projective module over $\mathbb{Z}[G]$. Suppose that the coinvariants of $P$ ...
6
votes
0answers
148 views

How to prove that a projective module is not free?

Let $A$ be a noncommutative (perhaps $\ast$-) algebra (over $\mathbb{C}$) and let $M$ be a projective module defined via a projector $P\in M_n(A)$; i.e. $M=P(A^n)$. Furthermore, assume that all ...
2
votes
1answer
84 views

Locally nilpotent operators of the Weyl algebra

$\newcommand{\ad}{\operatorname{ad}}$As my recent post (here) did not receive any answers yet, I thought I would ask a similar question in which I'm also interested. Let $A=$ $^{k \langle x,y\rangle ...
2
votes
0answers
62 views

Torsionfree finitely generated compact Iwasawa module

The following fact falls under the category of Iwasawa modules. Let $M$ be a torsion free finitely generated module over the non commutative noetherian ring $\Bbb{Z}_p[[G]]$, (where $G$ is a p ...
4
votes
1answer
144 views

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

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

Counting models in first order logics without existencial quantifiers

My question is about the posibility of to construct a parameter space of models in a first order theory, finitely presented, with out existencial quantifiers (parameter space in the sense of ...
1
vote
0answers
39 views

About alternating polynomials an the Rowen's notation

Some definitions... Definition 1: A polynomial $f(X_1,\dots ,X_d)$ is $t$-linear if the variables $X_1,\dots ,X_t,\; t\leq d$ appear in all monomials of $f$ and degree of $X_i,\; i=1,2,\dots ,t$ on ...
3
votes
0answers
69 views

Projective dimension of ring over its center

If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite? (Assuming that $A\neq Z(A)$).
6
votes
1answer
146 views

Is the following module over a group ring necessarily infinitely generated?

Suppose $\Gamma$ is a (finitely presented, but this is probably irrelevant) group, and $M$ is a finitely generated (EDIT: finitely presented) module over $\mathbb{Q}\Gamma$ which is ...
1
vote
3answers
340 views

smooth connected affine scheme over Z has good reduction almost everywhere

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a ...
1
vote
0answers
41 views

extend derivations of ore domain to its quotient field

I wonder whether someone knows a good reference(textbook or paper) for the following result: Any derivation of ore domain may be extended unqiuely to a derivation of its quotient field. Thanks.
3
votes
1answer
154 views

What do epimorphisms in noncommutative rings look like?

The question I want to ask is inspired by this mathoverflow post about epimorphisms in the category of commutative rings. I found the seminar (by P. Samuel) referenced by David Rydh particularly ...
0
votes
0answers
88 views

Relationships between finiteness of stable rank and IBN property of rings

Does any ring of finite stable rank have IBN property? Where can we find this result?
2
votes
1answer
92 views

Injective modules over noncommutative noetherian rings

Let $R$ be a left noetherian ring, then it is well known that Matlis proved that any injective $R$-module decomposes into a sum of indecomposable injectives, each of form $E(R/I)$ for some irreducible ...
4
votes
0answers
245 views

Reconstruction of noncommutative scheme

It is known that a quasi compact scheme(even quasi separated scheme)can be determined uniquely by the category of quasi coherent sheaves on it by Gabriel-Rosenberg reconstruction theorem The ...
4
votes
2answers
251 views

Maximal centralizer in full matrix ring

I will be so thankful if someone can help me with the following question. Is it possible to obtain all maximal centralizers in the full matrix ring, $M_n(F)$, for an arbitrary finite field $F$? Here, ...
5
votes
2answers
258 views

Polynomial identities for mod p matrices

Can there be a polynomial over the field $F_p$ of $p$ elements ($p$ prime) in non-commuting variables $X_1,..., X_r$ such that: 1) $f(A_1,...,A_r)=0$ for every $n \times n$ matrices $A_1,...,A_r$ ...
1
vote
1answer
160 views

What are the upperbounds of the Nil radical?

The main radicals of a non-commutative ring (with 1) are the Sum of all nilpotent ideals $\subseteq$ Prime radical $\subseteq$ Nil radical $\subseteq$ Jacobson radical $\subseteq$ Brown-McCoy radical. ...
0
votes
1answer
169 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 ...
1
vote
0answers
55 views

Does Castelnuovo-Mumford regularity hold for this $\mathbb{C}$-algebra$?

Let $R$ be a noncommutative finitely generated $\mathbb{C}$-algebra such that its center $S$ is smooth (in commutative sense) and $R$ is finite over $S$. Is there Castelnuovo-Mumford regularity ...
5
votes
5answers
747 views

Noncommutative computational package

I am wondering if there is a program which can do simple operations over noncommutative rings, like expand products and substitute one expression for another. To clarify, consider the following ...
15
votes
1answer
3k views

Conceptual explanation of Strassen's trick for matrix multiplication

Algorithms for fast multiplication of polynomials and integers have well-known conceptual explanations. A good survey paper is Daniel J. Bernstein's Fast Multidigit Multiplication for Mathematicians. ...
11
votes
2answers
603 views

A graded ring $R$ is graded-local iff $R_0$ is a local ring?

I asked this question some months ago on math.stackexchange.com: http://math.stackexchange.com/questions/126810/a-graded-ring-r-is-graded-local-iff-r-0-is-a-local-ring It would be great (for me) to ...
8
votes
5answers
775 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 ...
35
votes
1answer
996 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 ...