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21 votes
4 answers
7k views

Binomial Expansion for non-commutative setting

What could be a reference about binomial expansions for non-commutative elements? Specifically, where can I find a closed formula for the expansion of $(A+B)^n$ where $[A,B]=C$ and $[C,A]=[C,B]=0$? ...
Binai's user avatar
  • 829
21 votes
1 answer
2k views

Is there any non-commutative ring such that every element other than the identity is a zero divisor?

A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring"...
Salvo Tringali's user avatar
12 votes
0 answers
542 views

Does Wedderburn's Little Theorem hold constructively?

Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative. The proofs that I am aware ...
Martin Brandenburg's user avatar
8 votes
1 answer
685 views

The state of the art on topological rings - the Jacobson topology

I was recently studying the Jacobson density theorem and I found it quite interesting. Most textbooks I've seen, including Jacobson's own Basic Algebra, only spend a few lines about the reason why it ...
Melanzio's user avatar
  • 183
7 votes
1 answer
281 views

Question concerning the coefficients of block idempotents

Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$. Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$. For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...
Bernhard Boehmler's user avatar
7 votes
1 answer
179 views

Symmetry of unique generator property

In this article: Canfell, M. J. "Completion of Diagrams by Automorphisms and Bass′ First Stable Range Condition." Journal of algebra 176.2 (1995): 480-503. the author defines a ring $R$ to ...
rschwieb's user avatar
  • 1,507
6 votes
2 answers
449 views

Survey of recent developments of the Gelfand-Kirillov dimension

It is almost two decades since the now classical books by McConnell and Robinson's [ Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in ...
jg1896's user avatar
  • 3,318
6 votes
1 answer
256 views

Is there any structural characterization of the rings in which every element other than the identity is a (two-sided) zero divisor?

[I fear that I'm missing something obvious here, but I'll dare to ask anyway.] As we all know, a division ring is a (unital, associative, non-zero) ring where every non-zero element is a unit. So, let ...
Salvo Tringali's user avatar
6 votes
1 answer
936 views

Noncommutative HKR theorem

What is the analog of HKR theorem in the noncommutative world? Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a smooth commutative algebra $A$ of finite type over a ...
Sasha Patotski's user avatar
6 votes
1 answer
372 views

Cohn localization examples

I'm working on my master's thesis, part of which involves an exposition on Cohn localization. (nlab discussion) In Free ideal rings and localization in general rings, Sec 7.4, Cohn gives a ...
Tyler's user avatar
  • 161
5 votes
3 answers
2k views

Ideal structure of a tensor product of certain algebras

I would be grateful if anyone could give me a reference regarding the following question. Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar ...
Ilja's user avatar
  • 423
5 votes
1 answer
911 views

Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...
Kaveh's user avatar
  • 493
5 votes
1 answer
301 views

Discriminants of Clifford algebras

I have a Clifford algebra defined over a field of characteristic not equal to $2$. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...
John Palmieri's user avatar
5 votes
0 answers
442 views

A reference on semisimple linear algebra

Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings? In fact this question is a ...
Alexander Shamov's user avatar
2 votes
2 answers
105 views

Every non-zero submodule of $R_R$ has an indecomposable direct summand: True when $R$ is von Neumann regular?

Let's say that a (right) module $M$ is well complemented if every non-zero submodule of $M$ has an indecomposable direct summand (by the way, is there a better or more standard name for this property?)...
Salvo Tringali's user avatar
2 votes
1 answer
287 views

Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable

Crossposted on StackExchange on July 28 (no answer so far). Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can ...
Salvo Tringali's user avatar
2 votes
1 answer
192 views

Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR

Let $H$ be a (commutative or non-commutative) monoid. We say that $H$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ...
Salvo Tringali's user avatar
2 votes
1 answer
164 views

Algorithmically finite-dimensional (noncommutative) algebras.

Can anyone help to find some information about these structures?
Ayrana Mongush's user avatar
2 votes
0 answers
98 views

Do $r(a) \leq^\oplus R$ and $r(a) = r(a^2)$ imply $r(a) = eR$ and $aR \subseteq (1-e)R$ for some idempotent $e$?

Let $R$ be a (commutative or non-commutative, associative) ring with unity, and let $a$ be an element of $R$ such that $r(a) = r(a^2)$, where $r(\cdot)$ denotes a right annihilator. It follows that $r(...
Salvo Tringali's user avatar
1 vote
2 answers
333 views

Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
Rahul Sarkar's user avatar
1 vote
0 answers
39 views

Rings where every indecomposable principal right ideal is extensive

Let $R$ be a (commutative or non-commutative, associative) unital ring. Following Nicholson & Yousif [1, p. 21], we say that a right ideal $\mathfrak i$ of $R$ is extensive if every $R$-linear ...
Salvo Tringali's user avatar
1 vote
0 answers
124 views

On the rings $R$ with the property that $eR \cong fR$ for all primitive idempotents $e, f \in R$

Let $R$ be a (commutative or non-commutative) ring with identity. As usual, an idempotent $e \in R$ is primitive if $eR$ (the principal right ideal generated by $e$) is indecomposable as a right ...
Salvo Tringali's user avatar
0 votes
1 answer
72 views

Reference request: Left $R/k$-modules [closed]

In the paper titled: On the module of differentials of a noncommutative algebra and symmetric biderivations of a semiprime algebra I found the following definition: Let $k$ be a commutative ring with ...
The Student's user avatar
0 votes
1 answer
96 views

$\mathrm{rk}_R M$ vs $\mathrm{rk}_S M$ - how nice need $R,S$ be?

Let $R\hookrightarrow S$ be Noetherian (noncommutative) rings without zero divisors with $\mathrm{rk}_{R} S < \infty$ (e.g. $S=R*G$ the crossed product of $R$ with a finite group $G$). Let $M$ be a ...
Philip Kirschbaum's user avatar