Questions tagged [noncommutative-algebra]

For questions about rings which are not necessarily commutative and modules over such rings.

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2
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0answers
33 views

Subalgebra of $M_n(\mathbb{C})$ generated by two elements (along with unity)

Let $M_n(\mathbb{C})$ denote the algebra of $n\times n$ matrices over the field of complex numbers $\mathbb{C}$. Let $h_1,h_2\in M_n(\mathbb{C})$ be two Hermitian matrices. Suppose that $h_1,h_2$ are "...
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1answer
39 views

Induced module; what is the $K[G]$-action?

Given a finite group $G$, $H$ a subgroup, a field $K$, and a $K[H]$-module $M$, define $$\operatorname{ind}_H^G M := \lbrace f : G \to M : f(gh) = h^{-1}f(g)\ \text{for all}\ g \in G, h \in H\rbrace....
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34 views

What is the correct name for a “product function” on a monoid?

Let $W$ be a monoid. A function $f\colon W\rightarrow W$ is a "product function" if $f(w)$ is a product of constants in $W$ and positive integer powers of $w$. It could also be called a "non-...
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22 views

Dimension of the dual of a simple module over a simple $\mathbb{C}$-algebra.

I am given that A is a simple finite-dimensional associative unital algebra over the $\mathbb{C}$, and $M$ is a simple $A$-module. Furthermore, $V_M = \text{Hom}_A(M, A)$ is a right A-module with the ...
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1answer
17 views

Show there exists a non-commutative cancellative semigroup with generators $w,x$ satisfying $xwx=ww$

Let $W$ be a cancellative semigroup and $w,x\in W$ non-identity elements such that $xwx=ww$ and $\{x,w\}$ generates $W$. Prove that $W$ is commutative or find a counterexample. (Note that $x=1,w=2,W=\...
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1answer
27 views

If in a commutative ring $R$ a maximal ideal is nilpotent, then $R$ is local [duplicate]

Let $R$ a commutative ring, and $M$ a maximal ideal of $R$. If there exists $n\in \mathbb{N}$ such that $M^n=0$, then $R$ is local. In general, I have proved that $R/M^n$ is local with a unique ...
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1answer
30 views

Determinant of a matrix for non-commutative products

Let's suppose we have a $3 \times 3$ matrix $A=(a_{ij})$ whose elements $a_{ij}$ are objects whose product is not commutative: $a_{ij}·a_{kl}\neq a_{kl}·a_{ij}$. Then, which would be the formula for ...
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Showing that two path algebra quotients aren't isomorphic

I've been working through Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras, and I've gotten stuck on Exercise III.8(c). The larger exercise has me consider the following quiver $...
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1answer
46 views

Is there a module over a noncommutative domain in which the torsion elements do not form a submodule?

In my advanced Algebra course, we were told to prove when $R$ is an integral domain and $M$ an $R$-module, $\textrm{Tor}(M)$ is a submodule of $M$ (here $\textrm{Tor}(M)$ denotes the torsion of $M$, ...
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1answer
26 views

Is any ring with a homomorphism out of a full matrix ring necessarily itself isomorphic to a full matrix ring?

The title says it all. But actually, one should be answering the following stronger question: Question: For any ring $R$ and any positive integer $n$, is the functor $R/\mathbf{Ring} \to M_n(R) /\...
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1answer
17 views

Irreducible characters of k[G] when k is not algebraically closed and char k divides order of G.

Let k[G] be the group algebra where char(k) divides |G| with G being a finite group. Assume k is not algebraically closed. How can one show that the characters associated with the irreducible ...
3
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1answer
61 views

If $\mathrm{Ext}^n_R (M,R/\mathrm{rad}(R))=0$ for all $n\geq 1$, then $\mathrm{Ext}^n_R (M,N)=0$ for all $n\geq 1$.

Let $R$ be a ring with $1$, which is Noetherian and Artinian as a left $R$-module. Let $M$ be a left $R$-module and let $N$ be a finitely generated $R$-module. I am asked to show that if $\mathrm{Ext}^...
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2answers
71 views

Uniqueness of Artin-Wedderburn decomposition

I am studying Artin-Wedderburn structural decomposition theorem for semisimple rings. I understand that it says that any semisimple ring, $R$ is isomorphic (as rings) to $M_{n_1}(D_1) \times M_{n_2}(...
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2answers
108 views

Finite rings $R$ in which $x^{25}=x$ holds

I want to classify finite rings $R$ in which $x^{25}=x$ for all $x\in R$. I know Jacobson's Theorem that if $x^n=x$ for all $x\in$ then $R$ is commutative. I don't know how to show the Theorem for ...
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0answers
22 views

a belongs to J(R) if and only if Ra is left quasi regular left ideal of R

$ a\,\in\,J(R)\, \, if\, \,and\, \,only\, \,if\, \,Ra\, \,is\, \,a\, \,left\, \, quasi\, \,regular\, \,left \, \,ideal\, \,of \, \,R $ Here $J(R)$ is the Jacobson radical of $R$ where $R$ is a non ...
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1answer
70 views

(Left) Noetherian domains and Torsion submodules

By a domain I mean a non trivial ring without any zero-divisors (not necessarily commutative). Let $R$ be a ring and $M$ be a left $R$-module. We say an element $m\in M$ is a torsion element iff ...
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0answers
48 views

Dimension of center of k[G]/rad k[G] where characteristic of k divides the order of G.

Let G be a finite group and consider k[G] where k is a field. In the scenario where char(k) divides |G|, how can one show that the dimension of Z(k[G]/rad k[G]) is strictly less than dimension of Z(k[...
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1answer
37 views

A question about a maximal ideal $M$ in a non-commutative ring $R$ having identity but without zero divisors and its quotient ring $R/M$.

Does every maximal ideal $M$ in a non-commutative ring $R$ having identity but without zero divisors make $R/M$ a division ring? The question is equivalent to "Does there exist a non-commutative and ...
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1answer
21 views

Is the tensor product of prime algebras prime?

Recall, that an an associative algebra with unit $A$ is prime if whenever $a,b\in A$ have the property that for all $r\in A$, $arb=0$ then either $a=0$ or $b=0$. It is the correct way to extend the ...
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3answers
552 views

Example of an algebra which is not isomorphic to its opposite

I was trying to solve an exercise (marked with a star) that asks to come up with an example of an algebra $A$ which is not isomorphic to $A^{\mathrm{op}}$. I thought at first that I just need to ...
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2answers
182 views

Let $x^2=y^2=1$ and $xy\neq yx$. There are $\binom{2n}{n}$ expressions of length $2n$ in $x$ and $y$ that are equal to $1$.

This question is motivated by this link. The statement is as follows. (Edit: Even if there are already two great answers, I would love to have a couple more answers. Especially, I would like to see ...
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0answers
82 views

Suppose every functor which preserves $\oplus$ arbitrarily is additive when restricted to fin. gen. proj. modules, then is $S$ stably finite?

(This is the culmination of my recent efforts to understand this problem, and it will be the last post of this type I think. I have a feeling that it might not allow for an answer that fits a forum ...
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2answers
65 views

Must $\operatorname{Soc}(R)^2 =\operatorname{Soc}(R)^3$ (for a ring $R$)?

I am struggling with the following problem. Show that the right socle of a ring $E := \operatorname{Soc}(R)$ has $E^2 = E^3$. I know that $E$ is a two sided ideal and so $E^3 \subset E^2$. I am also ...
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1answer
47 views

Is there a functor $F$ of left modules preserving $\oplus$ by arbitrary isomorphism, but its restriction to fin. gen. proj. modules isn't additive?

Are there rings $R$, $S$ and a functor $F:{_R\textbf{Mod}}\to{_S\textbf{Mod}}$ such that For all left $R$-modules $M,N$, we have $F(M\oplus N)\cong F(M)\oplus F(N)$ via an arbitrary isomorphism, and ...
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2answers
508 views

Why is the constant term of $(1+x+y+xy)^n$ equal to $\frac{1}{2}\binom{2n}{n}$?

If we define this: for any $x,y$ such that $x^2=y^2=1,xy\neq yx$, express in terms of $n$ the constant term of the expression $$f_{n}=(1+x+y+xy)^n\,.$$ I guess this result is $\dfrac{1}{2}\binom{...
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1answer
30 views

Tensor product of matrix algebras over field

I want to prove that $Mat_{n_1}(k) \otimes_k Mat_{n_2}(k) \cong Mat_{n_1n_2}(k) $ (as $k$-algebras) where $k$ is a field by checking the universal property. Namely, I need to inclusion $Mat_{n_{1,...
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1answer
38 views

Describing all the irreducible and f.g. completely reducible modules over $\mathbb{C}[x,y]$

I couldn't even find all the irreducible modules over $\mathbb{C}[x,y]$ yet. I know that modules over $\mathbb{C}[x,y]$ is the same as $\mathbb{C}$-vector spaces $V$ equipped with two commuting ...
2
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1answer
18 views

Can we make any conclusion on the numerators of elements in the fraction division ring $D(x)$?

Let $D$ be a division ring. Let $D[x]$ be a polynomial ring in a central indeterminate $x$. And let $D(x)$ be the quotient ring of fraction of $D[x]$. So, let $f,g \in D[x]$ such that $g \neq 0$ and $...
0
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1answer
33 views

Evaluating a commutator involving exponentials?

I am interested in evaluating the commutator $[e^V,e^W]$ where we know that $[V,W]=c$, and $V$ and $W$ are vector fields. I know that $e^V= 1 + V + \frac{V^2}{2}+\frac{V^3}{6}+\dots$, so I can see ...
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2answers
50 views

Must the product of a nil ideal with a minimal right ideal be 0?

Let $R$ be a ring (not necessarily with identity or commutative). Suppose that $K$ is a a nil ideal, and let $M$ be a minimal right ideal of $R$. Must it then follow that $MK = 0$? As $M$ is minimal ...
2
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1answer
25 views

Is there any difference between a maximal regular left ideal and a regular maximal left ideal?

I am aware of the lemma that every regular left ideal of a ring is contained in a maximal left ideal that is regular. But still things are not very clear. Any help will be appreciated.
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1answer
27 views

Noetherian modules and Noetherian rings

I want to show that if $R$ is a Noetherian ring then $Mat_n(R)$ is also a Noetherian ring. It is obvious that $Mat_n(R)$ is a finitely generated $R$-module. So $Mat_n(R)$ is a Noetherian R-module. ...
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0answers
10 views

Bimodule over division algebras

Let $E_1$, $E_2$ be finite-dimensional division algebras over $\mathbb{Q}$. Let $X$ be a left $E_1 \otimes_\mathbb{Q} E_2^{op}-$module. In other words, $X$ is an $E_1-E_2-$bimodule, and $\mathbb{Q}$ ...
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0answers
15 views

If the monoidal categories of $R$-bimodules and $S$-bimodules are monoidally equivalent, must $R$ and $S$ be Morita equivalent?

If $R$ and $S$ are Morita equivalent rings, then the monoidal categories of $R$-bimodules and $S$-bimodules are monoidally equivalent. Now, consider the converse: Question: If the monoidal ...
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0answers
47 views

General formula for permutation of modes in Virasoro algebra/ Explicit formula for general OPE structure constants

Let $Vir_c$ denote the state space of the Virasoro VOA with central charge $c$. Let $n$ be a positive integer(i.e. $\geq 0$) and let $(n_k,...,n_2)$ be a sequence of positive integers. I am interested ...
6
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1answer
52 views

Commutativity up to scalar implies commutativity in an algebra

Let $A$ be a (not necessarily commutative) algebra over a field $k$. Suppose that for all $a,b\in A$, we have $kab=kba$, i.e. commutativity up to scalar. Show that then $A$ is commutative. In the ...
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0answers
17 views

Brauer Groups - Exponents

Let A and B be finite-dimensional central simple k-algebras. Let ${F/k}$ be a finite field extension. Prove the following fact: $$ {[A]=[B]}\ \ \Rightarrow\ {exp⁡(A)=exp⁡(B)} $$ $$\\$$ Now, I found ...
2
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1answer
44 views

Kronecker product for matrices over noncommutative fields

Suppose $A, B, C, D$ are matrices over a commutative field such that $AC$ and $BD$ are well defined. Then we know that $$(A \otimes B)(C \otimes D) = (AC) \otimes (BD).$$ Are there clean formulas ...
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0answers
10 views

Reduced norm of a central simple algebra

Let $F$ be a field, $K/F$ a finite field extension and $D$ a central simple algebra over $F$. More precisely, this means that $D$ is a simple $F$-algebra with its center $F$, and $\dim _F D < \...
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0answers
22 views

Suppose that there exists $q \in Q_{ml}(R)$ such that $ D(x) = x^{\sigma}q - qx^{*}$ for all $x \in R$. Then $q \in Q_{ms}(R)$.

My question is the following Lemma. Throughout this question, $R$ always denotes a prime ring with involution ${*}$ and an automorphism $\sigma$, which is not commutative. An additive map $D: R\...
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0answers
131 views

Reference for anti-commutative Binomial Theorem

Let $x,y$ be two elements of a ring satisfying $$xy=-yx.$$ Let $n \geq 0$. Then we can calculate $(x+y)^n$ as follows: If $n$ is even, then $$(x+y)^n = \sum_{0 \leq k \leq n,\, k \text{ even}} \binom{...
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1answer
43 views

what if matrix multiplications were commutative [duplicate]

Would that make things easier in any science if matrix multiplication were commutative? I mean are researchers working to find more exceptions to the general rule of matrix multiplication being not ...
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0answers
30 views

Is the quotient ring $R/I$ always Dedekind-finite, where $I$ is the two-sided ideal generated by all elements of the form $xy-1$ where $yx=1$?

I wonder whether the quotient ring $R/I$ is always Dedekind-finite for any ring $R$ if $I$ is the two-sided ideal of $R$ generated by all elements of the form $xy-1$ where $yx=1$. One might think ...
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1answer
34 views

In a non-commutative ring (possibly without identity) with no nontrivial automorphisms, do nilpotent elements form an ideal?

In an old exam appeared this statement: True/False: "Let $R$ be a ring with the property that the unique ring automorphism is the identity. Then the set of all nilpotent elements form an ideal". I'...
8
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1answer
162 views

Is there any example of a simple Abelian ring which is not domain?

A ring $R$ is called: simple if it has no two-sided ideal; a domain if it has no zero divisor; abelian if each idempotent of $R$ is central. Is there any example of a simple abelian ring which is ...
0
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0answers
16 views

Rotational Symmetry Groups of Tetrahedron and Hexagonal Plate

Firstly, I am confused on how to show clearly how the Tetrahedron ($T$) and Hexagonal Plate ($H$) are not abelian (commutative). And as a follow on, how would I find $i)$ a pair of elements of $H$, ...
1
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1answer
79 views

$(x,yz,t) = (x,y,t)z + (x,z,t)y$ holds in every Jordan algebra.

The identity $$(x,yz,t) = (x,y,t)z + (x,z,t)y$$ holds in every Jordan algebra. Remember that a Jordan algebra satisfies $xy=yx$ and $(x^2,y,x) = 0$ for all $x,y$. Here $(a,b,c) = (ab)c - a(bc)$ is ...
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0answers
34 views

In free special Jordan algebra is valid $T(x,y,z,t) = \frac{1}{4}([x,z] \circ [t,y] + [x,t] \circ [z,y])$.

In the free special Jordan algebra $SJ[X]$ is valid the equality $$T(x,y,z,t) = \frac{1}{4}([x,z] \circ [t,y] + [x,t] \circ [z,y]),$$ where $T(x,y,z,t) = (xy,z,t) - x(y,z,t) - y(x,z,t)$. Here $[x,y] ...
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0answers
21 views

Matrix over commutative ring with noncommutative diagonal perturbation

I'm interested in the inverse and determinant properties of a relatively small matrix $A+D$, where $A\in M_n(R_c)$ for a commutative ring $R_c$ and $D\in M_n(R_n)$ is a diagonal matrix over a non-...
0
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0answers
58 views

Why can't we derive a joint probability distribution for non-commuting random variables?

This is related to this post on physics stack exchange. Disclaimer: Note that I am a physicist and don't have much of a technical background and in particular do not have a good background in measure ...

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