Questions tagged [noncommutative-algebra]
For questions about rings which are not necessarily commutative and modules over such rings.
1,138
questions
2
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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 "...
2
votes
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....
1
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0answers
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-...
1
vote
0answers
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 ...
1
vote
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=\...
0
votes
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 ...
1
vote
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 ...
1
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0answers
34 views
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 $...
3
votes
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$, ...
1
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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) /\...
0
votes
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
votes
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}^...
1
vote
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}(...
9
votes
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 ...
1
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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 ...
2
votes
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 ...
4
votes
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[...
1
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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 ...
1
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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 ...
8
votes
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 ...
4
votes
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 ...
1
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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 ...
3
votes
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 ...
1
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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 ...
19
votes
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{...
1
vote
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,...
0
votes
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
votes
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
votes
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 ...
1
vote
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
votes
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.
0
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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.
...
1
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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}$ ...
2
votes
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 ...
2
votes
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
votes
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 ...
0
votes
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
votes
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 ...
0
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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\...
4
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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
$$(x+y)^n = \sum_{0 \leq k \leq n,\, k \text{ even}} \binom{...
-2
votes
1answer
43 views
what if matrix multiplications were commutative [duplicate]
Would that make things easier in any science if matrix multiplication were commutative?
I mean are researchers working to find more exceptions to the general rule of matrix multiplication being not ...
4
votes
0answers
30 views
Is the quotient ring $R/I$ always Dedekind-finite, where $I$ is the two-sided ideal generated by all elements of the form $xy-1$ where $yx=1$?
I wonder whether the quotient ring $R/I$ is always Dedekind-finite for any ring $R$ if $I$ is the two-sided ideal of $R$ generated by all elements of the form $xy-1$ where $yx=1$.
One might think ...
1
vote
1answer
34 views
In a non-commutative ring (possibly without identity) with no nontrivial automorphisms, do nilpotent elements form an ideal?
In an old exam appeared this statement:
True/False: "Let $R$ be a ring with the property that the unique ring automorphism is the identity. Then the set of all nilpotent elements form an ideal".
I'...
8
votes
1answer
162 views
Is there any example of a simple Abelian ring which is not domain?
A ring $R$ is called:
simple if it has no two-sided ideal;
a domain if it has no zero divisor;
abelian if each idempotent of $R$ is central.
Is there any example of a simple abelian ring which is ...
0
votes
0answers
16 views
Rotational Symmetry Groups of Tetrahedron and Hexagonal Plate
Firstly, I am confused on how to show clearly how the Tetrahedron ($T$) and Hexagonal Plate ($H$) are not abelian (commutative).
And as a follow on, how would I find
$i)$ a pair of elements of $H$,
...
1
vote
1answer
79 views
$(x,yz,t) = (x,y,t)z + (x,z,t)y$ holds in every Jordan algebra.
The identity
$$(x,yz,t) = (x,y,t)z + (x,z,t)y$$
holds in every Jordan algebra. Remember that a Jordan algebra satisfies $xy=yx$ and $(x^2,y,x) = 0$ for all $x,y$. Here $(a,b,c) = (ab)c - a(bc)$ is ...
1
vote
0answers
34 views
In free special Jordan algebra is valid $T(x,y,z,t) = \frac{1}{4}([x,z] \circ [t,y] + [x,t] \circ [z,y])$.
In the free special Jordan algebra $SJ[X]$ is valid the equality
$$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] ...
1
vote
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
votes
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 ...
