Questions tagged [noncommutative-algebra]
For questions about rings which are not necessarily commutative and modules over such rings.
1,224
questions
-4
votes
0answers
17 views
Non-commutative Archimedean-ordered ring without an unit [closed]
Give an example of a non-commutative Archimedean-ordered ring without an unit.
0
votes
0answers
19 views
Direct sum of simple modules is cyclic [duplicate]
Let $R$ a ring, and $M$, $N$ simple $R$-modules non isomorph. Prove that $M\oplus N$ is cyclic.
I am confused. If $(m,n)\neq 0$, $\phi: r\mapsto (m,n)r$ and the projections $\pi_i$ are R-homomorphism. ...
3
votes
0answers
48 views
Do Artin rings contain many commutative Artin subrings?
Suppose $R$ is a Artin ring and $x\in R$. Is there necessarily a commutative subring of $R$ which is also Artin and contains $x$? What if I want it to contain finitely many commuting elements $x_1,\...
2
votes
1answer
51 views
Loewy decomposition of differential operators
The paper by Fritz Schwarz, "Loewy decomposition of linear differential equations", contains the following lemma, which I try to prove in order to understand the algorithm which Schwarz ...
5
votes
0answers
98 views
Why does the exactness of a Koszul complex require commutativity?
Most references on Koszul complexes seem to assume that the elements $x_1,\ldots, x_n$ live in a commutative ring or are central. It appears to me that the proof also works provided the weaker ...
0
votes
1answer
33 views
What is the smallest dimension a non-commutative C*-algebra can have?
What is the smallest dimension a non-commutative C-star-algebra can have? Let $d$ denote this dimension. Clearly, $d\leq 4$ as $M_{2}(\mathbb{C})$ is a $4$-dimensional non-commutative C-star-algebra. ...
0
votes
0answers
18 views
Obtaining the product of any element with a Hamiltonian path in a Cayley graph [closed]
Consider the Cayley graph of a finite non abelian group, generated by a generating set, say $\{s,t\}$.
Let me explain my question by thinking of a small example. Let the Cayley graph have 5 vertices, $...
5
votes
2answers
121 views
Matrix exponential for non-commutative operator entries of matrix
I would like to find the matrix exponential $e^{iHt}$ of the Hermitian matrix $H$ where
$$
H=\begin{pmatrix}
\delta& \sqrt{2}a & 0\\
\sqrt{2}a^\dagger &0& \sqrt{2}a\\
0 &\sqrt{2}a^\...
0
votes
0answers
25 views
Finitistic dimension conjeture for $A^{op} $ implies the strong Nakayama conjecture for A
I have some trouble with some detail in the proof of the following theorem. Assume that the Finitistic dimension conjecture is true for $ A^{op} $ that is $ sup\{ proj.dim(M) \vert M \in mod(A^{op}) ~...
4
votes
0answers
94 views
Non-central tensor product of central algebras
If $K$ is a field, it is easy to show using a basis that if $A$ and $B$ are $K$-algebras, then $Z(A\otimes_K B) = Z(A)\otimes_K Z(B)$ (where $Z$ denotes the center). This is no longer true if we ...
1
vote
1answer
33 views
Do the matrix ring endofunctors have left adjoints?
Given any positive integer $n$, does the endofunctor $R \mapsto M_n(R)$ of the category of (unital) rings have a left adjoint?
One possible idea for constructing the left adjoint $L(R)$ at a ring $R$ ...
3
votes
0answers
26 views
Does Hochschild (co)homology preserve quasi-isomorphisms?
Does Hochschild (co)homology preserve quasi-isomorphisms? I.e. if we have an algebra in chain complexes $A$ and a chain complex $M$ that is a bimodule over $A$, we may form the cyclic simplicial ...
2
votes
1answer
57 views
Can we define “Algebra of fractions” like ring/field of fractions?
According to the answers to this question, if $R$ is a ring with no zero divisors, we can define a ring of fractions of $R$ if $R$ satisfies the Ore condition and that ring will be a skew-field. What ...
2
votes
1answer
73 views
BakerāCampbellāHausdorff formula for generators of $SO(3)$
By noting ${e_1 = (1, 0, 0), e_2 = (0, 1, 0), e_3 = (0, 0, 1)}$ and $E_i = [e_i]_\times$ the generators of $SO(3)$, then we have the following commutator properties :
$$
\begin{align}
E_1 &= [...
1
vote
1answer
30 views
The $q$ Multilinear Theorem
Let $R$ be the skew polynomial ring $k_\mathfrak{q}[x_1,\ldots,x_m]$ where $x_ix_j=qx_jx_i$ with $q\in k^*$ and for all $i<j$.
The $q$ Multinomial Theorem states that $$(x_1+\ldots+x_m)^r=\Sigma_{...
2
votes
0answers
84 views
Common multiples in a group ring
Consider the group ring $\mathbb{Q} F_2$ with rational coefficients over the free group in two generators ($a$ and $b$).
It is not an Ore-Domain: While it does not contain any zero-divisors, for ...
0
votes
0answers
31 views
Quaternion Algebras and zero divisors
I am going through a paper, https://www.sciencedirect.com/science/article/abs/pii/0196677488900144 and over there I found this statement in pg 2 confusing-
The structure of associative algebra over $\...
0
votes
1answer
12 views
Showing that $U(\mathfrak{sl}_2)$ is a coalgebra
We know that there is a coalgebra structure on $U(\mathfrak{sl}_2)$ as follows for any $z\in \mathfrak{sl}_2$:
$$\Delta(z)=1\otimes z+z\otimes 1, \qquad \epsilon(z)=0.$$
Can someone be so kind to ...
0
votes
1answer
96 views
“NONCOMMUTATIVE Algebra with a view towards Algebraic Geometry”?
Is there a noncommutative algebra book that is similar to Eisenbud's "Commutative Algebra with a view towards Algebraic Geometry" in the sense that fundamental and geometrically motivated ...
0
votes
0answers
16 views
Idempotent Laurent polynomial (in noncommuting variables)
Let $K$ be a field and $R=K\langle X_1,\dots,X_n,X_1^{-1},\dots,X_n^{-1}\rangle$ the Laurent polynomial ring in $n$ noncommuting variables. Can $R$ have idempotents distinct from $0$ and $1$?
2
votes
1answer
64 views
Generalized Nakayama's lemma over a non-commutative ring
Let $R$ be a ring, $J(R)$ its Jacobson radical, $M$ is a finite $R$-module. The following statement is usually called Nakayama's lemma: if $IM=M$ then $M=0$. This is true over any ring (commutative or ...
1
vote
1answer
29 views
Can I always write group elements $L,R$ as $VA$ and $VA^{-1}$ respectively, for some $V,A$?
My first question is, given two different elements $L,R$ in the Lie group $SU(N)$ can one always make write
$$L=V A, \quad \quad R=VA^\dagger,$$
for some $V, A \in SU(N)$ ?
Secondly, how does this ...
1
vote
1answer
106 views
Commutativity of a prime ring
Question :
Show that $R$ is a prime ring containing two commuting non-zero left ideals $I$ and $J$ $\implies$ $R$ is commutative, where "commuting idealsā means $ij=ji$ for all $i \in I$, $j \in ...
0
votes
0answers
44 views
Hardness of the conjugacy search problem
I have come across the following problem, which is considered as a mathematically hard problem to solve (one way trapdoor) used for cryptography.
Conjugacy search problem: Let $G$ be a non-abelian ...
4
votes
1answer
61 views
Does a finitely generated faithful module over an Artinian ring contain a regular element?
In the text
Nicholson -- Introduction to Abstract Algebra, 4th Ed (2012)
the claim of exercise $8(b)$ of exercise set $11.1$ is:
If $R$ is a left artinian ring with $1\ne 0$, and $M$ is a finitely ...
0
votes
0answers
27 views
On the Krull-Schmidt-Azumaya Theorem
$\newcommand\restr[2]{{% we make the whole thing an ordinary symbol
\left.\kern{-\nulldelimiterspace} % automatically resize the bar with \right
#1 % the function
\vphantom{\big|} % pretend it's ...
3
votes
1answer
38 views
Higher commutators in rings
In Herstein's "Noncommutative rings" it is proved that any rng $R$ such that for every $x,y\in R$ there exists an integer $n(x,y)>1$ such that $$(xy-yx)^{n(x,y)}=(xy-yx)$$ must be ...
1
vote
0answers
49 views
A simple module isomorphism in Wedderburn theory
Given a semisimple ring $R$, we have a $R$-module isomorphism $R \cong n_1V_1 \oplus n_2V_2 \oplus\cdots \oplus n_lV_l$ where $V_1, V_2, \cdots, V_l$ are nonisomorphic simple $R$-modules.
By ...
2
votes
1answer
41 views
Finding generalized inverses in the ring of linear transformations of a finite dimensional vector space V over a division ring
Let $L$ be the ring of linear transformations of a finite dimensional vector space $V$ over a division ring $D$. Show that for any $l$ belonging to $L$, then there exists a $u\in L$ such that $lul=l$. ...
3
votes
2answers
72 views
Why does the set of submodules of a module that are direct sums of irreducible submodules have a maximal element?
I was reading the proof that every R-module M of an artinian semisimple ring R is the direct sum of all irreducible R-submodules M. In the proof they stated that the set
$$\mathcal{F}:=\{N\subset M: N ...
1
vote
1answer
41 views
explicit formula of $(x \frac{\partial}{\partial x})^n f(x)$
Is there any explicit expression of
$$
(x \frac{\partial}{\partial x})^n f(x)
$$
as function of $x$ and $\frac{\partial^{k}f}{\partial x^k} $$, $$ 1\leq k \leq n$.
Any idea
Thanks
0
votes
0answers
9 views
resolutions over a Leibniz algebra
Given the usual definitions of Leibniz algebra $L$, ideal $I$ and module $M$, has the theory of resolutions e.g. of $L/I$ over $L$, been worked out?
citation please
1
vote
0answers
39 views
Is restriction of scalars preserving injective modules equivalent to flatness?
Given any ring homomorphism $R \to S$, if $S$ is a flat right $R$-module, then any injective left $S$-module is also injective as a left $R$-module.
Now, I'm wondering whether the converse is true. ...
1
vote
1answer
26 views
Locally nilpotent derivations closed under isomorphism?
I have been struggling to prove or find a counterexample to the following:
Fix a field $k$ of characteristic $0$. Let $R:= k[x_1,...,x_n]$ and $A$ be the ring of differential operators on $R$ (so $A$ ...
1
vote
1answer
53 views
Equality in non commutative field
Let $K$ be a non commutative field and $a,b \in K$ such that $ab \neq ba$.
Show that :
$$ a=\big[b-(a-1)^{-1}b(a-1)\big]\big[a^{-1}ba-(a-1)^{-1}b(a-1)\big]^{-1}. $$
I tried to move the second term of ...
0
votes
0answers
30 views
Jacobson radical and bimodules
I know that the Jacobson radical of a (non-commutative) rng $R$ is the intersection of all annihilators of right irreducible modules. I know also that one can replace "right" with "left&...
6
votes
1answer
41 views
An example about a non-commutative division ring with finite characteristic
After reading the proof of the theorem
āFor every central division $F$-algebra $D$ with $D$ $\neq$ $F$, $D$ contains a separable extension $K \supsetneqq F$ā,
I have a question: dose there exist a non-...
0
votes
0answers
38 views
Notation for functions of a Grassman variable
Let $V$ be a vector space over a field $\mathbb{K}$ and $\mathcal{O}(V)=\mathrm{Sym}V^*$ its ring of polynomial functions. Given $f_1,\dots,f_n\in V^*$, we have the homogeneous polynomial $F=f_1\cdots ...
2
votes
0answers
20 views
Fitting's Lemma version for pseudocompact modules or linearly compact modules
Let $R$ be a pseudocompact ring or a linearly compact ring. Is there a version of Fitting's Lemma for pseudocompoact or linearly compact $R$-modules?
0
votes
0answers
30 views
Taylor expansion for matrix exponential
Consider the matrix exponential
$$U(x)=e^{ix^j T^j}$$
where $T^j$ are matrices (in my particular application $U\in SU(4)$ and $T^j$ are its generators) and $x^j \in \mathbb{R}$. I would like to know ...
0
votes
0answers
49 views
Can you give me some concrete examples of varietes of some algebras?
Can you give me some concrete examples of varieties of some algebras?
I have seen many definitions for varietes of algebras, but I can not find concrete examples for this(for example, varietes of ...
4
votes
1answer
65 views
Infinite intersection of finitely generated ideals in a coherent ring.
It has been claimed without proof in several answers that an intersection of two finitely generated ideals in a coherent ring is finitely generated. Thus, the finitely generated ideals in a coherent ...
2
votes
2answers
72 views
Existence of Central Nilpotent Element implies that the Ring is not semi-simple
Show that if the ring $R$ with $1$ has a central nilpotent element then it is not semisimple.
I couldn't find a solution directly but I have a solution. Since any central nilpotent element is ...
1
vote
1answer
37 views
Definition of commutative and non-commutative algebra and algebra isomorphism
I am not sure of the meaning of the notation C<<...>> used to define a commutative algebra A and non commutative algebra A^ in the image attached. I do understand the meaning of the ideal. ...
0
votes
0answers
10 views
Unimodular rows over a polynomial ring in two variables
Let $D$ be a division ring that is not a field and suppose $\alpha, \beta \in D$ with $\alpha\beta\neq \beta\alpha$. Show that $\sigma=(x+\alpha, y+\beta)$ is a unimodular row over $R=D[x,y]$.
So I ...
0
votes
0answers
48 views
Example of a non-commutative ring containing 4 elements
I need an example of a non-commutative ring containing 4 elements. I considered the following example of $M_2(Z_2)$ containing the following four elements:
$\begin{bmatrix}0&0\\0&0\end{bmatrix}...
1
vote
0answers
79 views
Constructing an isomorphism between vector spaces
Let $A$ be a central simple $k$-algebra of dimension $n^2$ over $k$. Also assume that $A= {\rm End} (V)$, where $V$ is an $n$-dimensional $k$-vector space. For $0\leq i \leq n$, let $M_i$ be the set ...
1
vote
1answer
58 views
What is the tensor product dependent on the field in an algebra?
I don't understand the following notation:
$$V_F := V \otimes_k F$$
First of all, I know that that the product is a bilinear operation, i.e. $A \otimes A \to A$, between elements of the vector space $...
1
vote
1answer
48 views
An example of Artinian ring with an ideal K, where K is not Artinian ring?
It is true that an ideal of semisimple ring is semisimple ring. It is true that an ideal of an Artinian semisimple ring is an Artinian ring.
What about just Artinian? Is there an example with the ...
3
votes
2answers
71 views
Morita equivalence and Brauer equivalence
Let $k$ be a field and $A,B$ be two (finite-dimensional) central simple $k$-algebras. We usually say that $A$ and $B$ are Brauer equivalent (or similar) if their underlying division algebras (given by ...
