Questions about rings that are not necessarily commutative.

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### Attaching an ideal whose square is zero: does this operation have a name and a notation?

I know I met the following construction somewhere, but I cannot remember where. Let $A$ be
a (unital associative) ring, and let $N$ be an $A$-$A$ bimodule. On the product set $A\times N$ we define ...

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35 views

### Computing intersection of Weyl algebra ideal with certain subring

Let $D=k [x_1,\ldots, x_n, \partial_1,\ldots, \partial_n] $ be the nth Weyl algebra over the characteristic zero field $k $. Set $\theta_i=x_i\partial_i $. Let $I $ be a left ideal in $D $. Is there a ...

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**1**answer

74 views

### Maximal commutative subrings of the endomorphism ring of a vector space

Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...

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**1**answer

498 views

### A paper by Y. Morita

The corresponding bibliographical details are:
Yoshihito Morita, Elementary proofs of the commutativity of rings satisfying $x^{n}=x$. Mem. Defense Acad. 18 (1978), no. 1, 1–24.
Does anybody here ...

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86 views

### Free algebra and ring of quotients [closed]

I am reading of example T.Y.Lam 'A first Course in Noncommutative Rings':
Let $R=\mathbb{Z}\langle x,y\rangle/(y^2,yx)$. To work with $R$, we shall confuse $x,y$ with their images in $R$. Thus,we ...

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57 views

### Completion of an algebra

Based on arXiv:math/9802041v1, there is a definition for $NC$-filtration and $NC$-completion of an associated algebra over the complex numbers:
Let $R$ be an associative algebra and $R^{\rm Lie} = ...

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47 views

### Modification of nonfree locus

Let $ R $ be a commutative noetherian ring with identity. Let $ M $ be an $ R $-module. By definition the nonfree locus $ NF(M) $ of $ M $ is defined as the set of prime ideals $ {\mathfrak p} $ of $ ...

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85 views

### Ergodicity of elementary symmetric polynomials with noncommutable variables

Let $\{X_n\}$ be an ergodic sequence of random variables, $X_n:(\Omega,\mathcal{F})\to (S,\mathcal{S})$ where the target set $S$ is a matrix ring. My question is,
Can the following limit be found ...

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44 views

### In a Noetherian domain, what can one say if all non-zero non-units are in all but finitely many maximal right ideals?

Let $R$ be a (not necessarily commutative) Noetherian domain. Clearly, units of $R$ are not in any maximal right ideal, and elements of $J$, the Jacobson radical, are in all maximal right ideals. All ...

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167 views

### Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...

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95 views

### Quaternion algebra in characteristic $p$

Given a prime number $p$, can you give me concrete examples of fields $\mathbf F$ of characteristic $p$ and quaternion algebras $\mathbb H(\mathbf F)$ over $\mathbf F$ such that $\mathbb H(\mathbf F)$ ...

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86 views

### Differentially closed fields

Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$.
Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...

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**1**answer

146 views

### Existence of a skew field with surjective inner derivations

In my research, I've come twice now towards a skew field $K$ that satisfies the following:
$$\text{for all non-central element $a$, the map }\quad x\mapsto ax-xa\quad\text{ is onto.}$$
I am hoping ...

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105 views

### Quotient modules of polynomial rings by maximal one-sided ideal

Let $R[X]$ be a ring of polynomials over an associative unital ring $R$ which is not necessarily commutative. Let $M$ be a maximal left ideal in $R[X]$. It is easy to see that if the intersection of ...

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273 views

### Topological loops vs. algebro-geometric suspension in Hochschild homology

Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...

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125 views

### Rank versus free-rank of a module

Suppose M is a finitelly generated left module over a ring R. We define the rank of M as the minimal number of generators of M. If in addition M is free, then we define the free-rank of M as minimal ...

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336 views

### Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...

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163 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 ...

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223 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 $P$ ...

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161 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 ...

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112 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 ...

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68 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 ...

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150 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 ...

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79 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 ...

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43 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 ...

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77 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 $A\neq Z(A)$).

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156 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 ...

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369 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 ...

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96 views

### Examples of noncommutative Bezout domains

I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...

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48 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.

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174 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 ...

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92 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?

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**1**answer

116 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 ...

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268 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 ...

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308 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, ...

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275 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 $A_1,...,A_r$ ...

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162 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.
...

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179 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 ...

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57 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 ...

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893 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 ...

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**1**answer

4k 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.
...

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672 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 ...

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941 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 ...

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**1**answer

1k 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 ...