# Questions tagged [noncommutative-rings]

Questions about rings that are not necessarily commutative.

157
questions

**1**

vote

**1**answer

28 views

### If the Grothendieck ring of a semiring on a free commutative monoid is unital, is the original semiring unital?

Suppose $S$ is an associative semiring whose underling commutative monoid is free (in particular, cancellative) and that its Grothendieck ring $G(S)$ is a unital ring. Can we conclude that $S$ must be ...

**1**

vote

**1**answer

63 views

### If, in a unit-regular ring, the right annihilator of $a$ equals the right annihilator of $b$, then $aR = bR$?

Recall that a (unital) ring $R$ is von Neumann regular (VNR) if, for each $x \in R$, there exists $y \in R$ such that $x = xyx$; and unit-regular if such an element $y$ can be taken to be a unit.
...

**9**

votes

**1**answer

99 views

### Special nilpotent elements

Let $R$ be a (noncommutative, associative) ring. Set $N_2:=\{x\in R : x^2=0\}$, the set of nilpotent elements of degree $2$ (also called the square-zero elements).
If $x,y\in R$ satisfy $xy=0$, then $...

**5**

votes

**0**answers

131 views

### Representation theory terminology question

For a paper I'm writing, I need a term for a representation-theoretic concept that I'm sure someone has thought of before, so I thought I'd ask here rather than just make something up.
Let $G$ be a ...

**3**

votes

**1**answer

190 views

### $H^1(X, GL(n, \mathcal{O}_X))$ and Vector Bundles

Let $X$ be a nice space: a manifold, a variety over a field, or so on. We know that line bundles on $X$ up to isomorphisms are given by $H^1(X,\mathcal{O}_X^*)$.
Can it be generalized to higher rankal ...

**0**

votes

**1**answer

211 views

### Given a unitary commutative ring $R$, what are the rings $R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy)$ called

We are studying the rings
$$
R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right)
$$
Do you know if they have a name?

**1**

vote

**1**answer

127 views

### Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?

Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...

**4**

votes

**0**answers

141 views

### Do you know rings without involutions, auto-anti-isomorphics? In that case, what is the minimal example?

Do you know rings without involutions, but auto-anti-isomorphic (isomorphic to their opposite)? In that case, what is the minimal example?
If a ring has an involution f, then f is an anti-automorphism;...

**5**

votes

**1**answer

64 views

### Reference for a certain derivation on the ring of ordered series over a free monoid

Let $R$ be a (commutative or non-commutative) unital ring, $X$ be a non-empty set, and $R \langle\! \langle X \rangle\! \rangle$ be the ordered series ring (in fact, a ring of formal power series over ...

**1**

vote

**2**answers

210 views

### How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there?

How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there?
As much as I have searched, I have not found any results that answer my question; not even for k = 1,2.

**2**

votes

**1**answer

64 views

### Separability of $\mathbb{C}[x,y_1,\ldots,y_r]$ over $\mathbb{C} + (h,y_1,\ldots,y_r)$

The answer to this MO question says the following:
Lemma 1. Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unramified if and only if $h$ ...

**7**

votes

**2**answers

285 views

### Idempotent Laurent polynomials (in noncommuting variables)

Let $K$ be a field and $R=K\langle X_1,\dots,X_n,X_1^{-1},\dots,X_n^{-1}\rangle$ the Laurent polynomial ring in $n$ noncommuting variables. Can $R$ have idempotents distinct from $0$ and $1$?

**1**

vote

**0**answers

153 views

### Road map: beyond Artin-Wedderburn theorem

For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in ...

**3**

votes

**0**answers

38 views

### Reference for NIM-rep theory for non-commutative fusion rings?

The literature on nonnegative integer matrix representations (NIM-reps) seems to be focused on commutative fusion rings, since a primary motivation there is for rational conformal field theory (RCFT). ...

**5**

votes

**1**answer

356 views

### Do you know which is the minimal local ring that is not isomorphic to its opposite?

The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.

**4**

votes

**1**answer

67 views

### What is the extended centroid of a free algebra?

For a prime ring $R$, you can define its "Martindale ring of quotients" $Q(R)$. See for example:
Martindale, Wallace S. III, Prime rings satisfying a generalized polynomial identity, J. ...

**6**

votes

**1**answer

271 views

### Algebra with a certain abelian group as the multiplicative group

Let $A$ be an abelian group. Are there an algebra $\mathfrak{X}(A)$ s.t. the multiplication group is isomorphic to A ? i.e.
$$
\mathfrak{X}(A)^{\times} \simeq A.
$$
For example, for $A=\mathbb{Z}/4\...

**1**

vote

**0**answers

33 views

### Monomials in right ideal

First of all, hello everyone and thanks in advance of any kind of help.
Let $X =\{x_1,\dots,x_n\}$ and denote by $K\langle X\rangle$ the free algebra over some field $K$. Let $I = (f_1,\dots,f_m) \...

**2**

votes

**1**answer

180 views

### Endomorphism rings of infinitely generated free modules generated by idempotents?

Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...

**0**

votes

**0**answers

39 views

### Can Q(R) embed to Q((R ⊗ S )/ P)

Let $R, S$ be Noetherian $k$-algebra, where $k$ is a field, and $P \otimes S$ is Noetherian.
let $P$ be a prime ideal of $R \otimes S$ such that $P \cap (R \otimes 1) = (0) = P \cap (1 \otimes S)$, ...

**5**

votes

**1**answer

140 views

### $R/I\cong R/\text{Ann}_R(R/I)$ but $I\neq\text{Ann}_R(R/I)$

I originally asked this on Stack Exchange but with no luck. Upon doing research with some noncommutative rings, I thought of a curious question. Does there exist a noncommutative unital ring $R$ and ...

**0**

votes

**0**answers

73 views

### Gelfand–Kirillov dimension of the first Weyl algebra by using the definition

$\DeclareMathOperator\GKdim{GKdim}$Here I am trying to find the Gelfand–Kirillov dimension of the first Weyl algebra just by using the definition of the Gelfand–Kirillov dimension.
Let $A$ be an ...

**11**

votes

**3**answers

746 views

### Does Morita theory hint higher modules for noncommutative ring?

Two possibly noncommutative rings are called Morita equivalent if their left-module categories are equivalent. In the commutative case, Morita equivalence is nothing more than ring isomorphism. ...

**4**

votes

**1**answer

199 views

### Example of a projective module with non-superfluous radical

Let $R$ be a ring with unit. A submodule $N$ of an $R$-module $M$ is called superfluous if the only sumbodule $T$ of $M$ for which $N+T = M$ is $M$ itself.
It is shown, for example, in
[1] F. W....

**4**

votes

**1**answer

179 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 ...

**6**

votes

**2**answers

260 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 ...

**2**

votes

**0**answers

49 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**

votes

**1**answer

164 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 ...

**2**

votes

**0**answers

92 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**

votes

**1**answer

208 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**

votes

**1**answer

207 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 ...

**7**

votes

**3**answers

540 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 $C &...

**2**

votes

**0**answers

46 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 ...

**3**

votes

**1**answer

95 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 $Ra+Rb=R$, ...

**3**

votes

**2**answers

230 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 ...

**2**

votes

**1**answer

128 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**

votes

**0**answers

125 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 ...

**3**

votes

**0**answers

59 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 ...

**3**

votes

**1**answer

82 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 ...

**1**

vote

**1**answer

480 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 ($...

**3**

votes

**0**answers

86 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 $yx-xy=1$.
...

**2**

votes

**0**answers

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 ...

**4**

votes

**1**answer

206 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

**1**answer

156 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 ...

**2**

votes

**1**answer

120 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 ...

**2**

votes

**0**answers

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 $R$ ...

**4**

votes

**1**answer

296 views

### Possible values of symmetric functions evaluated on quaternions

$\DeclareMathOperator\sym{sym}$Let $i$, $j$, $k$ be 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 ...

**1**

vote

**0**answers

51 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 ...

**5**

votes

**0**answers

326 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 $...

**1**

vote

**0**answers

68 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 $R$ ...