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Questions tagged [noncommutative-algebra]

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

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Bound on dimension of centralizer of algebras

First I will write the problem: Assume $A$ is a simple ring (has no non trivial two sided ideals) and is left Artinian, denote $F$ to be its center, and $B \subseteq A$ a subring containing $F$, if $...
Amir Mg's user avatar
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49 views

A generalization of center of a ring

Let $R$ be a ring. The center of $R$ denoted by $Z(R)$ is defined as $Z(R)=\lbrace x \in R | xy=yx~\forall y\in R\rbrace$. Are there any generalizations of center of ring? I know one structure but I ...
Chaudhary's user avatar
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1 answer
37 views

Construct an isomorphism between $A$ and $M_2(F)$ for $4$-dimensional central simple algebra $A$ over $F$ with nonzero zero divisor

Let $A$ be a $4$-dimensional central simple algebra over $F$ with nonzero zero divisor, how to construct isomorphism between them? My goal is to construct an explicit isomorphism without Wedderburn's ...
Samuil Lee's user avatar
-1 votes
1 answer
51 views

Maximal subfield of division ring [closed]

I see some book says if $D$ is a division ring, $K$ is the maximal subfield of $D$, then $K=C_{D}(K),$ where $C_{D}K:=${$d\in D|dk=kd,\forall k\in K$}. But if division ring $D$ is given then how to ...
T100's user avatar
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20 views

Tensor of left Artinian algebras need not be left Artinian.

We have the following fact: If $A$ and $B$ are left Artinian algebras over a field $K$, then $A\otimes_{K}B$ need not be left Artinian. But, how can I actually constuct the example satisfied the ...
T100's user avatar
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1 vote
1 answer
56 views

proof of two simple left R module must be isomorphic

Theorem: If $R$ is simple left artinian,then any two simple left $R$ module is isomorphic. pf: since $R$ is simple left artinian, so by Wedderburn artin theorem, it is isomorphic to the matrix algebra ...
lee's user avatar
  • 405
2 votes
1 answer
22 views

Von Neumann regular rings and its ideals

Let R be a strongly regular Von Neumann ring, this is, that for every $r \in R$ there exists $x \in R$ such that $r^2x=r$. From here, how to prove that $R$ is strongly Von Neumann regular if and only ...
Apopip's user avatar
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1 answer
41 views

If R is semisimple left Artinian, then R is its own quotient ring.

I am trying to do the following exercises in Hungerford. If $R$ is semisimple left Artinian, then $R$ is its own quotient ring. By the Wedderburn-Artin theorem we know that $R$ is isomorphic to sum of ...
T100's user avatar
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1 answer
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Prime radical and Jacobson radical

Let $R$ be a ring (not necessary commutative and not necessary contains 1) In Hungerford, it define $P(R)$ is the prime radical, that is the intersection of all prime ideal of $R$. And, $J(R)$ is the ...
T100's user avatar
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4 votes
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Prime radical of a ring

Let $\mathbb{Z}$ be the ring of integers and $a,b,c \in \mathbb{Z}$. Consider $R= \Bigg\{\begin{pmatrix} a & c\\ 0 & b \\ \end{pmatrix} \big| a-b\equiv c\equiv 0\mod2 \Bigg\}$. ...
Chaudhary's user avatar
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0 answers
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semisimple module and jacobson radical

Exercise from Hungerford IX 3.8: Let $A$ be a (possibly nonunitary) module over a left Artinian ring $R$ such that $Ra \neq 0$ for all nonzero $a \in A$, and let $J = J(R)$. Then if $J(R)A = 0$ then $...
lee's user avatar
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1 vote
1 answer
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Finding the projective cover of a quotient of ring matrices.

Let $R$ be the ring $\begin{pmatrix} \mathbb{Z}_{2} & \mathbb{Z}_{2} \\ 0 & \mathbb{Z}_{2} \end{pmatrix}$ and $I= \begin{pmatrix} \mathbb{Z}_{2} & \mathbb{Z}_{2} \\ 0 & 0 \end{...
Cos's user avatar
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2 votes
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20 views

A reference for the definition of a minimal polynomial when ring is noncommutative

Let $R$ be a non-commutative ring and consider $R$ as a subring of $M_n (R)$ the ring of $n\times n$ matrices with entries in $R$. Let $A\in M_n (R)$. I was wondeing if someone could give a reference ...
Mahtab's user avatar
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prove that the column vector is left quasi regular.

I want to prove that $M_n(J(R))\subset J(M_nR)$: the following is my approach: let $R$ be a ring, $M_1=[m_{ij}]\in Mat_nJ(R)$ where $J(R)$ is jacobson radical of $R$, I want to show two fact: $1.$if $...
lee's user avatar
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3 votes
1 answer
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Is a surjective endomorphism of a finitely generated module over a *non-commutative* ring with unity necessarily an isomorphism?

As is well known, any surjective endomorphism of a finitely generated module $M$ over a commutative ring with unity $R$ must be an isomorphism. What about the non-commutative case? In other words, is ...
Liang Chen's user avatar
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2 answers
94 views

Technical question about computing the left ideals of the ring $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$

Computing the left ideals of $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$. This is the same that computing the ideals of $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$, ...
Cos's user avatar
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7 votes
1 answer
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Show a ring is commutative if $r^2 = r + r$

The claim is that any ring $R$ in which for all $r \in R$ we have that $rr = r + r$, must be commutative. No assumptions are made about $R$ having multiplicative identity or being commutative. I was ...
Andrew E's user avatar
-1 votes
0 answers
105 views

Compute the left ideal of the upper triangular matrix with entries in $\mathbb{Z}_{2}$. [duplicate]

An apparently similar question has been asked in Is there a nice way to classify the ideals of the ring of lower triangular matrices? But it's not the same since the those are not upper triangular ...
Cos's user avatar
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0 votes
2 answers
67 views

coproduct(free product) of algebras is free iff both factors are free algebra?

Coproduct of associative algebra is defined in wiki, let us work on field so that coproduct exist generally. And for free algebra I mean semigroup ring generated by semi free groups on $n$ alphabet ...
wer's user avatar
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1 vote
1 answer
61 views

Can there be a subgroup $K$ of a group $G$ such that for some $a \in G$, $aK \subseteq Ka$ but $Ka \nsubseteq aK$? [duplicate]

I have a question, in a sense, about how asymmetric left and right cosets can be when dealing with an infinite, non-normal subgroup $K$ of a (non-abelian) group $G$. Specifically, my question is ...
MathNeophyte's user avatar
1 vote
1 answer
46 views

Artin-Wedderburn application - finding simple modules

I'm trying to solve the following: Let $R$ be the $\mathbb{R}$-algebra $R:=M_2(\mathbb{R})\times M_3(\mathbb{H})\times\mathbb{C}\times\mathbb{C}$. Determine how many simple (left) $R$-modules there ...
quanticbolt's user avatar
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1 answer
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When will $\mathbb{Z}$ be a projective $\mathbb{Z}G$-module?

Let $G$ be a finite group and $\mathbb{Z}G$ be the integral group ring. I want to know for what groups $G$ we have the result. It seems we should focus on the tautological short exact sequence for $\...
Rellw's user avatar
  • 13
0 votes
0 answers
30 views

Endomorphisms of torsion points of Drinfeld modules

A Drinfeld module is defined to be an $\mathbb F_q$-algebra morphism $\phi: \mathbb F_q[T] \rightarrow K\{\tau\}$, where $K=\mathbb F_{q^m}$ is a finite field and $K\{\tau\}$ is defined as the ring of ...
Reyx_0's user avatar
  • 1,128
0 votes
1 answer
24 views

Pointwise multiplication of modules homomorphism in noncommutative case

Let $A$ be a noncommutative (unital associative) ring and $M,N$ be left $A$-modules. It's well known that $\operatorname{Hom}_A(M,N)$ is an Abelian group with respect to the pointwise addition. But ...
Mitya Kustov's user avatar
1 vote
3 answers
81 views

An algebra as the generator of a non-commutative ring

I am using an older book, Advanced Quantum Theory by Paul Roman (1964), to study the Dirac equation. Roman makes a lot of references to some terms in abstract algebra that I am not completely ...
Nada Band's user avatar
1 vote
1 answer
78 views

Is generator of minimal left ideal is necessarily (semi-)idempotent?

Let $A = \mathbb{C}[G]$ be finite group algebra. Assume $I = Ax$ is a minimal left ideal (has no non-trivial left ideal properly contained in it) for some element $x \in A$. Is it necessarily true ...
SimB4's user avatar
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0 votes
1 answer
28 views

Division rings Jacobson’s commutativity theorem [closed]

I have a question, if you can help me, related to this theorem: I don't quite understand what I need to demonstrate and what is meant by $n(a, b)$. Let D be a division ring such that , for any $a,b \...
Miriam's user avatar
  • 11
2 votes
1 answer
69 views

Is there always an automorphism distinct from identity in a simple module?

Let $M$ be a simple module over a unital ring $R$. If $|M| \leq 2$, then $M$ has only one automorphism (the identity). I'm wondering whether for $|M| > 2$, there always exists an automorphism ...
Tom's user avatar
  • 1,158
0 votes
0 answers
54 views

How to prove when $R$ is not a commutative ring, $M\otimes_R N$ is not a module

I know how to prove when $R$ is a commutative ring, $M\otimes_R N$ is a module, but how to prove when $R$ is non-commutative ring, we cannot give any module structure on $M\otimes_R N$? I see many ...
MGIO's user avatar
  • 117
3 votes
0 answers
71 views

Using Bergman's Diamond Lemma to prove a PBW theorem

I've found myself in a situation where I need to prove a PBW theorem for a certain quotient of an algebra $A/I$ where $A$ is a filtered algebra given by generators and relations. Furthermore $A$ has a ...
LT1918's user avatar
  • 334
1 vote
0 answers
33 views

Two algebra structures on endomorphisms

Let $(\mathcal{M}, \otimes, \mathbb{k})$ be a symmetric closed monoidal category, which in my application is the category of $dg$-modules over some commutative ring. Let $A$ be a bialgebra/bimonoid in ...
Brendan Murphy's user avatar
2 votes
0 answers
43 views

Lie algebra cohomology of formal non-commutative vector fields

Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
Qwert's user avatar
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1 vote
1 answer
48 views

To what extent are submodules of $R^n$, for a ring $R$, describable by systems of linear equations?

Let $R$ be a ring and consider $R^n$ as an $R$-module. I'm wondering under what circumstances is a submodule of $R^n$ the solution set of some homogeneous system of linear equations in $n$ variables, ...
Tom's user avatar
  • 1,158
1 vote
1 answer
29 views

Does this property generalize to modules over noncommutative rings?

Let $M$ be a finite simple module over a ring $R$. For commutative $R$, it is easy to show that for every $r \in R$, the function $M \rightarrow M; x \mapsto rx$ is either bijective or constantly $0$. ...
Tom's user avatar
  • 1,158
1 vote
1 answer
46 views

Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
Brendan Murphy's user avatar
2 votes
0 answers
79 views

Non commutative Non reduced ring

Let $R$ be a ring. $R$ may not be reduced and commutative. Is it true in general ring that if for any $a,b,c\in R$, we have $abc=bca=cab=0$ then it implies $acb=0$. We can prove this in case of ...
Chaudhary's user avatar
  • 335
0 votes
1 answer
83 views

Inverse of a 2x2 quaternion matrix: whence this formula?

From here, the inverse of a $2\times2$ quaternion matrix is given by $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} =\frac{1}{\mathrm{Nrd}} \begin{bmatrix} |d|^2\overline{a}-...
coiso's user avatar
  • 3,151
0 votes
1 answer
39 views

Non commutative field of characteristic $\neq 0$ [closed]

Does there exist a non-commutative field of a characteristic different from zero?
Wannes De Maeyer's user avatar
2 votes
0 answers
23 views

Is the Eigen spectrum of a matrix completely defined by the algebra of its parts?

Consider two vector spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, where $0<n<m<\infty$. Now, I'd like to define matrices $A\in\mathbb{C}^{n\times n}$ and $B\in\mathbb{C}^{m\times m}$ in the ...
Jun_Gitef17's user avatar
1 vote
2 answers
68 views

In a ring $1+\epsilon+\cdots+\epsilon^{p-1}=0$ implies $1+\epsilon^c+\epsilon^{2c}+\cdots+\epsilon^{(p-1)c}=0$ for $0<c<p$.

I am a bit unsure about my following "proof", in particular the rationality of "taking projection". Problem: $R$ is a ring with identity, not necessarily commutative. If $\epsilon\...
Asigan's user avatar
  • 1,759
2 votes
1 answer
66 views

Fixed points of automorphism over quaternions

I know that for any $y\in\mathbb{H}\setminus\{0\}$, the map $\rho_y:\mathbb{H}\to\mathbb{H}$, $\rho_y(x):=y\overline{x}y^{-1}$ is an automorphism. I would like to know, for fixed $y$, which are the ...
Giulio Binosi's user avatar
2 votes
1 answer
99 views

Form of ideal generated by subset of noncommutative nonunital ring

Given a noncommutative nonunital ring $R$ and a subset $S\subseteq R$, I know that the left and right ideals generated by $S$ in $R$ have the forms$$RS:=\{\sum^n_{i=1} r_is_i:n\in\mathbb{N}\backslash\{...
Ancillary's user avatar
1 vote
1 answer
73 views

Finite Group's Group Algebra over a Field is a Principal Ideal Ring?

A (left) principal ideal ring (PIR for short) is a ring such that for every left-ideal $I$, there exists a $a \in R$ such that $I=Ra$. Firstly, similar to the case of PIDs, is that a ring $R$ is a PIR ...
SalutaFungo's user avatar
1 vote
0 answers
49 views

An identity in the generalized quaternion algebra

The generalized quaternion algebra $D$ is defined relative to a field $F$ of characteristic $2$ with parameters $a, b \in F^\times$ by $$D = F\langle i, j \rangle/(i^2 = a, j^2 = b, ij = - ji).$$ This ...
Brendan Murphy's user avatar
1 vote
0 answers
20 views

Noncommutative Nakayama's Lemma for Maximal Ideal Inclusion

In "Grobner Bases and the Computation of Group Cohomology" Hypothesis 1.5 is: Let $k$ be a field of characteristic $p$. Let $\Lambda$ be a finite dimensional $k$-algebra (associative with ...
user2154420's user avatar
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0 votes
0 answers
22 views

A normal regular element in a connected graded ring $S$ can become central by a suitable Zhang twist

I am reading the article Noncommutative quadric surfaces (here) by S. Paul Smith and M. Van den Bergh. There is a sentence in the introduction to the article: $S$ is a not-necessarily-commutative ...
Well's user avatar
  • 380
1 vote
0 answers
94 views

Derivations in a central simple algebra

In Farb's Noncommutative Algebra, the following exercise is given. Theorem 3.22 Let $R$ be a finite dimensional central simple $k$-algebra. Every $k$-linear derivation on $R$ is inner. I solved the ...
the_dude's user avatar
  • 512
2 votes
1 answer
169 views

Is $f(\operatorname{rad} A ) \subseteq \operatorname{rad} B$ when $f$ is a not surjective $K$-algebras homomorphism and $K$ is a field?

Let $K$ be a field. For a $K$-algebra $A$ take the definition of the Jacobson radical of $A$ as the intersection of all maximal left ideals of $A$. If $A$ and $B$ are two finite dimensional $K$-...
Hector Blandin's user avatar
0 votes
0 answers
41 views

Trace of iterated commutators and binomial coefficients

Let $A$ and $B$ be two matrices. I am trying to prove the following formula (and also find the conditions on $A$ and $B$ for it to work, if it is not true for any $A$ and $B$) : $$\underset{n=0}{\sum^...
Denis _J's user avatar
0 votes
1 answer
64 views

$f(m)=0$ for every $f\in \text{Hom}(M,S)$ then $m\in \text{rad}(M)$

Let $M,S$ be two right $A$-modules, the latter simple. Assume that for a certain $m\in M$ we have that $f(m)=0$ for every $f\in \text{Hom}(M,S)$. I want to show that then $m\in \text{rad}(M)$, where ...
kubo's user avatar
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