# Questions tagged [noncommutative-rings]

Questions about rings that are not necessarily commutative.

227
questions

3
votes

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### Tensor product and idempotents

Given an associative unital ring $R$ and an idempotent $e^2=e\in R$ one knows that $\operatorname{End}_R(Re)\simeq eRe \simeq \operatorname{End}_R(eR)$ as rings. Also we have a surjective $R-R$-...

6
votes

1
answer

366
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### On commutator of bounded operators

Let $\mathbb H$ be a Hilbert space and let $\mathcal B(\mathbb H)$ be the bounded operators on
$\mathbb H$. Let $J,K\in \mathcal B(\mathbb H)$ such that
$
J=J^*, K=-K^*.
$
Then the commutator $[J,K]$ ...

1
vote

1
answer

89
views

### Quotient rings of integral quaternion rings

I'm having a hard time finding information about the quotient rings of the Lipschitz quaternions and the Hurwitz quaternions.
The Lipschitz quaternions are defined as the quaternions with integral ...

1
vote

0
answers

52
views

### On a lemma of projective dimension

Let $R$ be a finite-dimensional algebra, and $A=R\oplus A_1\oplus A_2\oplus \dotsb$ be an $\mathbb{N}$-graded algebra which is locally finite (i.e. all $A_i$'s are of finite dimension). Let $\text{...

1
vote

0
answers

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views

### Is the matrix ring $\mathbb{M}_n(R)$, $n\geq 2$, over a serial ring $R$ again serial?

Let $R$ be a ring with $1$. A right $R$-module $M$ is called uniserial if its submodules form a chain, i.e., for any two submodules $A,B\subseteq M$ either $A\subseteq B$ or $B\subseteq A$. The module ...

10
votes

1
answer

207
views

### Matrix ring isomorphisms of different sizes

Do there exist (unital, associative, noncommutative) rings $R$ and $S$, where $\mathbb{M}_2(R)\cong \mathbb{M}_3(S)$, but these matrix rings are not isomorphic to $\mathbb{M}_6(T)$ for any ring $T$?

4
votes

1
answer

192
views

### Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?

All rings are assumed to be associative and have a 1. Let $k$ be a commutative artininan ring and $R$ a finitely generated $k$-algebra. Is it true that if $R$ is connected and hereditary, then $k$ is ...

1
vote

1
answer

239
views

### Wedderburn theorem for finite-dimensional algebras over the complex numbers

I'm trying to understand how to apply the Wedderburn theorem in the context of unitary algebras over $\mathbb{C}$ that are finite-dimensional and semisimple. Let $\mathcal{A}$ be a $\mathbb{C}$-...

1
vote

0
answers

52
views

### Ideals of Laurent polynomial ring over matrix ring

Let $K$ be a field. Let $R=M_2(K)\langle x,x^{-1}\rangle$ be the ring obtained from the matrix ring $M_2(K)$ by adjoining two elements $x$ and $x^{-1}$ which are inverse to each other ($x$ and $x^{-1}$...

2
votes

2
answers

180
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### Minimal ideals and subalgebras of semisimple algebras

I'm considering an algebra to be a ring which is also a vector space over some field $F$, and the algebra $A$ is said to be semisimple if it is semisimple as a ring, i.e., $A$ can be written as a ...

3
votes

0
answers

161
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### Amalgamation of commutative subrings

Let $A$ and $B$ be commutative subrings of a non-commutative ring $X$.
Is there always a commutative ring $Y$ containing $A$ and $B$ preserving their intersection?
This is equivalent to ask if in the ...

5
votes

1
answer

181
views

### Are module finite algebras over semiperfect rings again semiperfect?

Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...

3
votes

1
answer

232
views

### Hattori-Stallings trace

Let $R$ be a (possibly non-commutative) unital ring and $M$ be a left $R$-module. If $M$ is finitely generated and projective, the natural map $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...

2
votes

1
answer

109
views

### Local rings whose the endomorphism rings of E(R/J) is division ring

Let $R$ be a local ring with maximal ideal $J$. Assume that ${\rm End}_{R}({\rm E}(R/J))$ is a division ring (${\rm E}(R/J)$ means the injective envelope of $R/J$). Does $R/J$ is injective?

3
votes

1
answer

283
views

### Is every graded hereditary ring hereditary?

Let R be a graded (associative, unital) ring. If R is left graded hereditary (i.e. its left graded global dimension is 0 or 1), does it follow that R is left hereditary (i.e. its left global dimension ...

2
votes

0
answers

160
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### A direct proof that every projectivity between parallel lines is affine

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...

4
votes

1
answer

240
views

### Kaplansky inverse element theorem on group C-star algebra

In a class talking about $C^*$ algebra and (higher) index theory, I heard a theorem
(related to Kaplansky, proved?), that is
Suppose $\Gamma$ is a group (admitting Haar measure if necessary) while $\...

17
votes

1
answer

1k
views

### Why is the bicategory viewpoint useful?

In ring theory one often wants to think about bimodules as being morphisms between rings using tensor product as composition. However, this composition is only associative if one uses isomorphism ...

6
votes

1
answer

353
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### Morita equivalences and centers of some algebras

Let $k $ is an algebraically closed field of $\text{ch}(k)=0$.
Let $$E := k \langle x_0, x_1, x_2 ,x_3 \rangle/(x_ix_j-q_{ij}x_jx_i )_{0 \leq i,j \leq 3},$$ where $$(\text{deg}(x_0), \text{deg}(x_1), \...

2
votes

0
answers

136
views

### Zero divisors in the extra-special group algebra $\mathbb{R}[2^{1+6}_+]$

Can you characterize the unit-group of the real group-algebra of the extraspecial plus-type 2-group of order 128? (That is $\mathbb{R}[2_+^{1+6}]$ using Conway's notation.)
(Please choose any irrep ...

1
vote

0
answers

89
views

### Structure of the pre-Lie ring

I am having difficulty understanding the following result from an article:
Theorem: Let $(A, +, .)$ be a pre-Lie ring with
$(A, +) \cong \mathbb{Z}/p^3\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ ...

4
votes

0
answers

69
views

### Indecomposable injectives over Weyl algebras

Let $A=A_n(\mathbb{C})$ be the $n$-th Weyl algebra over the complex field. Then $A$ is a left Noetherian noncommutative ring. Is there a complete classification of indecomposable injective $A$-modules?...

6
votes

1
answer

408
views

### Ring in which $x^n-x$ is central for every $x$

Let $R$ be a ring , $n \gt 1$, such that for all $x \in R$: $x^n-x \in Z(R)$, the center of $R$. Does it follow that $R$ is commutative?
For $n=2,3$ this is pretty straightforward to prove. But what ...

1
vote

0
answers

33
views

### Number of right divisors of a central skew polynomial

Let $\mathbb{F}$ be a finite field of $p$ elements, $\sigma \in \operatorname{Aut}(F)$ of order $m$, $\mathbb{F}^\sigma$ be the fixed field of $\sigma$, and $\mathbb{F}[x,\sigma]$ be a skew polynomial ...

1
vote

1
answer

126
views

### Polynomial identities satisfied by the Weyl algebra in prime characteristic

The rank $n$ Weyl $A_n(\mathsf{k})$ algebra over a field $\mathsf{k}$ of zero characteristic does not satisfies any polinomial identity. If it were a PI-algebra, Kaplansky theorem would apply (since ...

1
vote

0
answers

99
views

### Does the center of any finitely generated associative algebra over a field have finite type?

Consider the monoid algebra $R:=K\langle x_1,\dots,x_n\rangle$ generated by $n$ letters $x_1,\dots,x_n$ for $n>1$ over field $K$. Equivalently, $R$ is the tensor algebra $T(V)$ on the $n$-...

3
votes

0
answers

89
views

### Cohn's localization for rings with enough idempotents

I am in the following situation: I have a non-unitary (associative) ring $R$ with enough idempotents or, if you prefer, a small pre-additive category. Actually, I even know that $R$ is right coherent (...

6
votes

0
answers

253
views

### Usefulness of total algebras and exotic generating series

In his first Algebra volume, Bourbaki [1] defines the structure of a “total algebra” i.e. the space of functions on a monoid $M$ (to a ring $k$) with the convolution product ( a function $f:\ M\to k$ ...

2
votes

1
answer

158
views

### Non-negative integer matrix representation of a fusion ring

Context: I am a physics grad student working on topological lines in 2D CFTs.
Let $A$ be a unital based $\mathbb{Z}_{+}$ ring with finite rank (or a Fusion ring) with the basis $B = \{b_1, b_2, \dotsc ...

2
votes

1
answer

154
views

### Relation(s) between units and nilpotent elements in graded noncommutative rings

In Commutative Algebra we have the following standard facts which I am going to state in a slightly different form than usually found in textbooks. Namely, let $A$ be a commutative unital ring of ...

5
votes

1
answer

330
views

### Simple component that is not a two-sided ideal

Suppose $R$ is a semisimple ring and if $L$ is a minimal left ideal. Let $B$ be the direct sum of all minimal left ideals isomorphic to $L$ ($B$ is called a simple component corresponding to $L$). It ...

3
votes

1
answer

84
views

### Topology of the Malcev-Neumann group ring

For a ring $R$ and a group $G$ the group ring $R[G]$ consist of maps from $G$ to $R$ with finite support.
It was shown that if the group is fully ordered them this ring can be embedded in a division ...

2
votes

1
answer

86
views

### Primitive group rings and endomorphism rings

It is known that, for any group $G$, there exists a group $H$ containing $G$ such that the group ring $F[H]$ for some field $F$ is primitive, see Formanek, Edward; Snider, Robert L., Primitive group ...

0
votes

0
answers

49
views

### Complemented subalgebra in a free Lie ring

A Lie ring is a triple $(G,+, [\ ,\ ]),$ where $(G,+)$ is an abelian group and $ [\ ,\ ]$ is a bilinear map satisfying
$[x,x]=0$
$[\ ,\ ]$ is bilinear
$[[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,...

7
votes

1
answer

430
views

### Center of a monoid ring

According to the Wikipedia page the center of a group ring $R[G]$ is the set:
$$
\{ p | \forall g,\, h \in G.\, p(g) = p(hgh^{-1}) \}
$$
i.e. class functions which do not distinguish elements of the ...

3
votes

1
answer

135
views

### Ideals of an ordered ring

Suppose $R$ is a strictly ordered (non-commutative) ring, in particular $ab > 0$ for any $a,\, b > 0$, that is also discrete in that there are no elements between $0$ and $1$.
Now consider a two-...

2
votes

0
answers

156
views

### Embedding a monoid into a group via its monoid ring

Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...

0
votes

1
answer

104
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### Do you know of any indecomposable ring that has no isolated elements and is neither reversible, nor integral, nor nilpotent, nor unitary?

Let $R$ be a non commutative ring. We will say that an element of $R$ is isolated if it is zero divisor and nothing nonzero annihilates it at the same time on both sides.
Note that there are many ...

3
votes

1
answer

324
views

### Graded global dimension of a graded algebra

Let $k$ be an algebraically closed field of characteristic $0$.
Let $A := k \langle x,x^{-1},y \rangle /(xy-qyx, x^{d_1}-ay^{d_2})$, where deg$(x)>0$, deg$(y)>0$, $q,a \in k^*$ and $d_1\text{deg}...

2
votes

0
answers

160
views

### Simple modules of quantum planes

Let $k$ be an algebraically closed field.
Let $R := k\langle x,y \rangle/(yx-qxy) (q \in k^*)$.
We often call $R$ a quantum plane.
If $q$ is a primitive $n$-th root, then for any $(\zeta, \xi) \in k^* ...

0
votes

0
answers

68
views

### countable direct sum of cyclic abelian $p^{2}$ groups

Let $G={{\Bbb{Z}}_{p^{2}}}^{(\aleph)}$ (countable direct sum of copies of ${\Bbb{Z}}_{p^2}$). It is clear that every subgroup of $G$ is a homomorphic image of $G$. Now this is my question:
Is it true ...

0
votes

1
answer

85
views

### Unital subrings of simple Artinian rings

Let $S=M_n(D)$, the ring of $n\times n$ matrices with entries in a division ring $D$. Now suppose that $R$ is a simple unital Artinian subring of $S$. Is it the case that $R\cong M_k(D')$ for some ...

6
votes

0
answers

231
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### Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra

This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...

4
votes

0
answers

88
views

### Nullstellensatz for maximal left ideals of quantum plane

Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...

2
votes

1
answer

277
views

### Primitive elements in the universal enveloping algebra of Lie superalgebra

Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...

6
votes

2
answers

356
views

### Abelian groups such that $A \cong \mathrm{End}(A)$ and "complete rings"

Motivation: for any ring $R$ there is the natural monomorphism $\mathrm{in} \colon R \to \mathrm{End}(R_{add}): r \mapsto (x \mapsto rx)$, where $R_{add}$ is an additive abelian group ( rings are ...

1
vote

0
answers

132
views

### Pseudo-coherent complexes over sheaves of non-commutative rings

I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion.
Assume that $\mathcal{R}_X$ is a ...

3
votes

0
answers

78
views

### Explicit separability idempotent for the center of a separable algebra

Let $A$ be a $k$-algebra for some commutative ring $k$. Recall that $A$ is said to be separable over $k$ if the multiplication map $A\otimes_k A^{\operatorname{op}}\to A$ has a section as a map of $A\...

4
votes

0
answers

152
views

### A non-commutative, left duo ring whose only unit is the identity

Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$.
Question. Is there a non-commutative, left ...

5
votes

1
answer

249
views

### Rings s.t. each element has a power lying in the center (and their completely prime ideals)

Let $R$ be a ring (throughout, all rings are associative and unital). We say $R$ satisfies condition (C) if, for every $a \in R$, there exists an integer $n \ge 1$ (depending on $a$) such that $a^n$ ...