Questions about rings that are not necessarily commutative.

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6
votes
1answer
130 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
294 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
37 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
129 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
76 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
73 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
213 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
181 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, ...
3
votes
1answer
195 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
153 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
155 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
52 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
522 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 ...
12
votes
1answer
2k 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
531 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 ...
4
votes
4answers
571 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 ...
34
votes
1answer
867 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 ...