# Questions tagged [noncommutative-rings]

Questions about rings that are not necessarily commutative.

134
questions

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### 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 ...

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**2**answers

205 views

### Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix

Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in
C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative ...

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44 views

### Existence of nontrivial transfinite divisibility in $R$-modules

Let $R$ be a unital, possibly noncommutative ring and $s \in R$. For a right $R$-module $M$, define $Ms = \{ms \mid m \in M\}$; this is an additive subgroup of $M$, which is a module over the ...

**7**

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148 views

### Categories of modules generated under coproducts by a small set?

Question 1: For which rings $R$ does there exist a small set $S \subseteq Mod_R$ such that every module $M \in Mod_R$ is a direct sum of modules in $S$?
Equivalenty, for which rings $R$ does there ...

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81 views

### Rings whose finitely-generated modules are co-hopfian

Let $A$ be a unital, possibly noncommutative ring. Dischinger showed [1] that the following are equivalent:
For every $a \in A$, there exists $n \in \mathbb N$ such that $a^n A = a^{n+1} A$;
For ...

**5**

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**1**answer

173 views

### Classification of finitely generated modules over non-commutative rings

Let $\Lambda$ be a commutative integral ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$...

**4**

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179 views

### Linear algebra over non-commutative semirings

I'm reading up on linear algebra over semirings, and I'm wondering why people seem to stop short of showing an equivalence between linear transformations between free modules and matrices.
It seems ...

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412 views

### What other lattices are obtainable from this noncommutative ring?

Here I will regard $SU(2)$ as the multiplicative group of unit quaternions.
There are just three conjugacy classes of finite subgroups $G < SU(2)$ where $[G:C] > 2$ for all cyclic subgroups $...

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44 views

### Non-singular rings which are Rickart

A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$.
It turns out that a ring $R$ is right Rickart iff every ...

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82 views

### Example of an associative unital ring R with stable range 1 and Jac(R)=0 that is not an exchange ring

Rings are supposed to be associative and unital, but not necessarily commutative.
Some definitions:
(Bass) A ring $R$ is said to have stable range $1$ if for all $a,b \in R$, whenever $Ra+Rb=R$, ...

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209 views

### Is Hilbert basis theorem true for positive graded ring?

Let $R=\oplus_{I\geq 0}R_i$ be a positive graded ring(maybe not commutative), where $R_0$ is a commutative Noetherian ring. If $R$ is finite generated $R_0$-algebra, is $R$ Noetherian?
In here, Is ...

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**1**answer

110 views

### Non-commutative projective lines

There have been many approaches to the notion of projective line:
combinatorial approaches (e.g. as certain permutation groups, such as $\mathrm{PGL}_2(k)$ in its natural action on $\mathbb{P}^1(k)$, ...

**3**

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86 views

### Constructing a centrally primitive idempotent in the group algebra of the symmetric group

Consider the group algebra of the symmetric group $ \mathbb{C} S_k$.
Given some Young tableau $T$ of shape $\lambda$, let $a_{\lambda,T}$ and $b_{\lambda,T}$ be the row symmetrizer and column ...

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55 views

### Ring epimorphisms and finiteness assumptions

Let $f : A\to B$ be a ring epimorphism. It is well-known that, under the extra assumption that $A$ and $B$ are commutative, then $f$ makes $B$ a finitely-generated $A$-module implies that $f$ must be ...

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77 views

### Irreducible skew polynomials over an algebraically closed field

Let $\mathbb{F}$ be a field, and denote with $\mathbb{F}[t,\sigma]$ the skew-polynomial ring, where $\sigma$ is an automorphim of $\mathbb{F}$. Recall that the multiplication of this ring is defined ...

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244 views

### A property for primitive idempotents

Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($...

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76 views

### Is a specific endomorphism of $A_1$ an automorphism?

Let $k$ be a field of characteristic zero, and let $A_1(k)$ be the first Weyl algebra, namely, the associative non-commutative $k$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$.
...

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40 views

### Partially commutative elements in powers of augmentation ideal

Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...

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180 views

### Noetherian ring with a “strange” idempotent ideal

Do you know a left-noetherian ring $R$ with a two-sided ideal $I$ such that:
$I=I.I$;
$I$ is not projective as a left $R$-module (and better, the tensor product over $R$ of $I$ with itself is not a ...

**6**

votes

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144 views

### Relative Dickson (trace) criterion for Jacobson radical?

In the following, all algebras are associative and unital. Let $J\left(A\right)$ denote the Jacobson radical of an arbitrary algebra $A$. Recall that this is defined as the set of all $a \in A$ such ...

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votes

**1**answer

102 views

### Field of definition of a finite dimensional division algebra and how to reduce it

Let F be a field, and E/F an infinite algebraic extension. Let D be a finite dimensional division algebra over E (meaning its center is also E).
Is it possible to somehow gow down to a finite ...

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89 views

### What are all pairs $(R,M)$ of a ring $R$ and a two-sided $R$-module $M$ such that all endomorphisms of $M$ are scalar multiples of $\text{id}_M$?

I was playing with some endomorphism rings and got curious whether there is a classification of all two-sided (not necessarily unitary on any side) modules $M$ over a (not necessarily unital) ring $R$ ...

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204 views

### Possible values of symmetric functions evaluated on quaternions

Let $i,j,k$ the units of quaternions, in particular $i^2=j^2=k^2=-1, ijk=-1$.
We will use non commutative variables $x,y,z$. Define $sym_{a,b,c}$ to be the polynomial made of the sum of monomials ...

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49 views

### Infinite Non Abelian Extensions Unramified Outside p

Let $K$ be a number field and $p$ be a fixed odd prime. Suppose $\mathfrak{p}\mid p$ is the only prime prime above $p$ in $K$, and that $p$ does not divide the class number of $K$ (I am okay with ...

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275 views

### Slightly noncommutative Nakayama's lemma?

Nakayama's lemma asserts the following. If $R $ is a commutative ring with an element $s$, and $M$ is a finitely generated module such that $sM = M$, then there exists $r \in R$ such that $rM =0$ and $...

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66 views

### Localizing prime ideals over Noetherian rings

Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ ...

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240 views

### Rings that fail to satisfy the strong rank condition

In T.Y. Lam's book Lectures on Modules and Rings, a ring $R$ is said to satisfy the strong rank condition if, for every natural number $n$, there is no right $R$-module monomorphism $R^{n+1}\to R^n$. ...

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47 views

### Single source shortest path over non-commutative finite idempotent semiring in Cartesian product

Let $G$ be a Cartesian product of two arbitrary directed weighted graphs $M$ and $N$.
The weights are from a non-commutative finite idempotent semiring.
Do there exist advanced results on the single ...

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61 views

### von Neumann regular ring homomorphisms

Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat.
In particular, $\mathrm{id}...

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97 views

### Do twisted group rings of free abelian groups admit universal fields of fractions?

Let $R$ be an associative ring with unit. Recall that an epic $R$-field is a ring epimorphism $\alpha\colon R\to D$ to a skew field/division ring $D$. An epic $R$-field $\alpha$ is called a field of ...

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202 views

### Independence of characters with respect to polynomials

I came across the following property :
Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors,
$\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\...

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107 views

### Localizations of group algebras of free groups

$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...

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39 views

### Relation between left projections

Let $A$ be a Baer *-ring. Let $x$ be in $A$, $L(x)$ is the left projection of $x$ that is the smallest projection with $L(x)x=x$.
Q. Let $p,q$ are projections in $A$ with $p\leq q$. For a given ...

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35 views

### The statue of a sequence of finite projections

Let $A$ be a Baer $*$-ring. Let $\{p_n\}$ be a sequence of finite projections in $A$. True or false?
Suppose that there is no $N$ with $p_n=p_{n+1}$ for $n\geq N$. We have then $\inf_{1\leq n\leq ...

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34 views

### something concerning finite projections

Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$).
Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$.
Q. Can we say that ...

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**1**answer

71 views

### Strongly finite projections in $*$-rings

Let $A$ be a $*$-ring. Let us have some points:
i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$.
ii) On the set of projections, we write $p\leq q$ if $pq=p$.
iii)...

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440 views

### Curious anti-commutative ring

Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context?
Here $\Lambda[x_0, x_1, x_2, \ldots]$ is the (graded-)commutative algebra (either ...

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227 views

### Is every (left) graded-Noetherian graded ring (left) Noetherian?

I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) ...

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92 views

### Extensions of modules of type $FP_n$

Let $R$ be a ring (not necessarily commutative, but with a unit). Recall that an $R$-module $M$ is of type $FP_n$ if $M$ has a partial projective resolution of length $n$ whose terms are all finitely ...

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142 views

### Algebraic version of unilateral shift

It was confirmed that Wold-type decomposition can be extended from von Neumann algebras to Baer*-rings (see this paper). By algebraic tools the notion of unilateral shifts is successfully transmitted ...

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195 views

### Torsion in a tensor product over a group ring

Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra.
Is it true ...

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211 views

### invertibility of matrix over free associative algebra

For a commutative ring $R$, a matrix $A \in M_n(R)$ is invertible iff $\det (A)$ is a unit in $R$. Is there a similar criterion to determine invertibility (having two-sided inverse) of a matrix over a ...

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169 views

### number of indecomposable summands of an extension of two modules

I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of ...

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**1**answer

359 views

### Looking for example of quotient of group algebra by ideal of group ring which fails to be injective

I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...

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44 views

### A question related to reflexive rings

Let $R$ be a ring. An endomorphism $\alpha:R\to R$ is said to be right central reflexive if for all $a,b\in R,$ $aRb=(0)\implies bR\alpha (a)\subset Z(R)$, where $Z(R)$ denotes the centre of the ring. ...

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84 views

### From socle of quotients to socle of ring itself

Let $I_1, \dots , I_n$ be ideals of a ring $R$ with identity having zero intersection. Assume that for some $x\in R$, $x+I_ i$ is an element of the right socle of $R/I_ i$, for each $ i=1,\dots , n$....

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65 views

### When is $R/Soc(R)$ reduced?

Let $R$ be a ring with identity. It is readily checked that when the quotient $R/S_r$ is reduced, the nilpotent elements of $R$ fall into $S_r$, where $S_r$ is the right socle of $R$. Is the converse ...

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98 views

### Nilpotent Elements and the Socle [closed]

Let $R$ be a ring with identity such that the quotient ring $R/S_r$ is abelian, i. e., all idempotents of the quotient are central. Here, $S_r$ means the right socle of the ring $R$. Do the nilpotent ...

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51 views

### An Abelan quotient ring by Socle

Let $R$ be a ring with identity whose (right) socle $S$ contains its nilpotent elements. Is it necessarily true that the quotient $R/S$ is an abelian ring? ( By an abelian ring I mean a ring whose ...

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108 views

### m-systems and n-systems in topological rings

Note that throughout rings have a multiplicative identity and are not necessarily commutative
Definition: Let $R$ be a ring and let $M\subseteq R$. Then, $M$ is an m-system iff for every $x,y\in ...