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

learn more… | top users | synonyms

2
votes
1answer
42 views

Constant term of noncommutative $(X+Y+(XY)^{-1})^n$

As the title reads I am trying to find a formula for the constant term of the above noncommutative polynomal expression, $$[1](X+Y+(XY)^{-1})^{3n}\quad \bigg(\in \mathbb{C}\langle X^{\pm 1},Y^{\pm ...
1
vote
0answers
18 views

If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.

Let $I$ be a left ideal in a semiprime ring $R$. If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.
0
votes
1answer
27 views

Let $R$ a left artinian ring. Show that if $a \in R$ is not a right zero divisor, then $a$ is a unit in $R$. [duplicate]

Let $R$ a left artinian ring. Show that if $a \in R$ is not a right zero divisor, then $a$ is a unit in $R$. Comments: I am tryed to do so: See the R-module $_{R}R$ and consider the function $f: ...
3
votes
2answers
67 views

Conceptual approach to the formula $\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)=\text{tr}(A)\det(v_1, \ldots, v_n)$

I answered this question earlier showing that $$\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)=\text{tr}(A)\det(v_1, \ldots, v_n),$$ and while I am happy with my answer, I feel like there should ...
4
votes
1answer
49 views

If $R/I$ and $R/J$ are semisimple, then so is $R/I\cap J$.

Let $R$ be a not necessarily commutative ring. If $I$ and $J$ are (two-sided) ideals in $R$ such that $R/I$ and $R/J$ are both semi-simple rings, then so is $R/I\cap J$. I tried the following: ...
0
votes
1answer
32 views

Simple Artinian ring $S$ is isomorphic to a matrix ring over a division ring?

I'm working on building up a proof of Artin-Wedderburn theorem, given in some of the exercises of Dummit and Foote. 18.2.9 says that if $S$ is a simple, unital ring, satisfying the DCC on left ...
1
vote
1answer
23 views

Matrix Ring of a Semisimple Ring

I recently read the concept of semi-simplicity of a (not necessarily commutative) ring. A ring $R$ is said to be semi-simple if $R$ as a left module over itself is a semi-simple module (This in turn ...
1
vote
0answers
15 views

On algebraicity of a formal power series

In his paper "Noncommutative identities" M. Kontsevich states the following: Theorem 2. For any $P=P(x,y)=1+\cdots\in\mathbb{C}[x,y]$ expand ...
1
vote
1answer
18 views

Are the isomorphism classes of simple left ideals in a semisimple ring finite?

Suppose $R$ is a unital semi simple ring, not necessarily commutative. It's known that there are only finitely many isomorphism classes of simple left ideals. Are these isomorphism classes ...
0
votes
1answer
337 views

Hom-tensor adjunction

Let $A$ be a ring (which might or might not be commutative), and let $M,N$ and $K$ be three bi-modules over $A$. There are two hom-tensor adjunctions. One says that $Hom_A(M\otimes_A N, K) \cong ...
1
vote
1answer
28 views

torsion free module and injective envelop of this

suppose that $R$ be a domain, $M$ a torsion free $R$-module and $V=E(M)$, the injective envelop of $M$. Is it true that if $M$ is torsion free then $V$ is torsion free? I guess it is true because $V$ ...
4
votes
1answer
103 views

An R-matrix in a quasitriangular Hopf algebra

I am new to the theory of Hopf algebra. So I am sorry if the following question has really trivial answer. Suppose that we have a quasi triangular Hopf algebra with the universal R-matrix $R$. It ...
2
votes
2answers
20 views

Simple generator modules

Let a ring $R$ with identity element be such that the category of left $R$-modules has a simple generator $T$. My question: "Is $T$ isomorphic with any simple left $R$-module $M$?" I tried the ...
0
votes
1answer
18 views

Show that $B^{(1)} = 0$ and $B^{(2)}$ have basis $\{[x_i,x_j]; i>j\}$

Definition: A polynomial $f \in K \langle X \rangle$ is called a proper polynomial, if it is a linear combination of products of commutators: $$f(x_1,x_2,...x_n) = \sum ...
0
votes
1answer
51 views

How can I maintain linear independence through a commutator?

Consider a Lie algebra $\mathcal{L}$, a linearly independent generating set $\mathcal{G}$, and an element $X \in \mathcal{L}$. Edit: Note that $\mathcal{G}$ is not necessarily a basis; the generation ...
2
votes
1answer
60 views

Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$?

Consider the ring $M_2(\mathbb{R})$ of all $2 × 2$ matrices over $\mathbb{R}$. Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$? We know, in the ring $\mathbb{Z}$, ...
2
votes
1answer
26 views

Proving that a Left semisimple ring $R$ is both left noetherian and left artinian

Prove that a left semi-simple ring $R$ is both left noetherian and left-artinian. I am following the proof given in pg 27,A first course in non-commutative rings (T.Y.Lam). Its strategy is to show ...
0
votes
0answers
19 views

Rentschler's theorem on non-commutative algebras

Rentschler's theorem says that every locally nilpotent derivation of the algebra $A=\mathbb{C}[x,y]$ (i.e., a linear map $\phi$ that satisfies the Leibniz rule and such that for every $p \in A$ ...
2
votes
0answers
32 views

How is this commutator property $[a,b]=bx$ called?

If two elements $a,b$ commute like this $[a,b]=bx$ for some $x$ so that it can rewriten as $$ ba=a(b-x), $$ is there a term for such property? It is as if $a$ and $b$ almost commuted (but not quite) ...
1
vote
1answer
37 views

Associated primes of an $R$-module

An associated prime of an $R$-module $M$ is an ideal of the form $Ann_R(N)$ where $N$ is a prime sub-module of $M$ in the sense that $N$ is nonzero and $Ann_R(N)=Ann_R(N')$ for each nonzero sub-module ...
0
votes
0answers
39 views

Simple algebra that is not a simple ring

maybe this question is trivial, however I'm not acquainted with non-commutative stuff. In http://www.encyclopediaofmath.org/index.php/Simple_algebra, it's written that a simple algebra may not be a ...
0
votes
1answer
46 views

How to find all ring structures over $C_2\times C_2$?

$C_2$ denotes the cyclic group of order 2. How to find all ring structures over $C_2\times C_2$? The question is equivalent to give a full list of all essentially different bilinear 2-operations ...
1
vote
0answers
22 views

If $R$ is a noncommuative Noetherian ring, is it true that $R[[x]]$ is a Noetherian ring?

I am looking for a reference in order to answer the following question: If $R$ is a noncommutative Noetherian ring, is it true that $R[[x]]$ is a Noetherian ring? The answer is well-known in the ...
0
votes
0answers
48 views

Fact on Iwasawa module

The following fact falls under the category of Iwasawa modules. Let $M$ be a torsion free finitely generated module over the non commutative noetherian ring $\Bbb{Z}_p[[G]]$, (where $G$ is a p ...
5
votes
0answers
76 views

NCG with all noncommutativity in a nilpotent ideal

While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative ...
1
vote
0answers
51 views

If $R$ is generated by idempotents, then $\text{Ann}(R)=0$?

Let $R$ be a ring (not necessarily commutative or unital) which is generated by idempotents. I'd like to know if $\text{Ann}(R)=0$ must holds. Here I use $\text{Ann}(R)$ to denote the set of all ...
1
vote
1answer
29 views

Finite dimensional central division algebras over a finite extension of $\mathbb{F}_q(T)$

Over number fields, finite dimensional central division algebras are always cyclic algebras. So the construction of cyclic algebras is a nice recipe to create algebras, which exhausts all finite ...
2
votes
1answer
29 views

Non-commutative noetherian integral domain-Ore condition

Let $R$ be a non-commutative integral domain with unity which is also a right Noetherian ring. By integral domain I mean that the product of nonzero elements is always nonzero. I am trying to show ...
0
votes
0answers
31 views

Isomorphism involving the opposite of a ring

Given a ring $R$, its opposite ring $R^{op}$ is defined as the ring formed by considering the same underlying set of $R$ with the same addition but with multiplication performed in the reversed order. ...
2
votes
1answer
30 views

Do any $\ell^{p}(\omega)$ have the extension property?

Definition 1. A metric space (normed space) $X$ has the extension property exactly in case, for all finite $A\subseteq X$ and isometric (linear isometric) $f:A\rightarrow X$, there exists an extension ...
3
votes
0answers
52 views

Prove that under some condition the commutator subgroup contains not only commutators

I'm trying to prove the following statement: Let's assume $ G $ is a finite group. Let $ Z(G) $ denote its center, $(G : Z(G))$ the index of $ Z(G) $ in $ G $ and $ [G; G] $ the commutator subgroup ...
1
vote
1answer
13 views

Equivalence of Semisimplicity

In Noncommutative algebra by Benson Farb, there is an exercise concerning this result; An R-module M is semisimple if every submodule of M is a direct summand. where semisimple is defined as M being ...
0
votes
0answers
33 views

An essential right ideal in a ring

Let $S⊆R$ be rings with unity such that $S_S$ is essential in $R_S$. If $r∈R$ is a nonzero element there exists an $s_0∈S$ with $rs_0$ a nonzero element of $S$. Now, could we find a right ideal $I$ ...
0
votes
0answers
36 views

On exponentials of formal power series

I am having a very hard time trying to understand the following paper by M. Kontsevich (http://arxiv.org/pdf/1109.2469v1.pdf), and since I cannot really find a way out by myself, I here to seek some ...
1
vote
2answers
40 views

The expression “commute to something”

I'm in a quantum mechanics class, where people started using expressions such as: "operators $\hat{x}$ and $\hat{p}$ commute to $i\hbar$", to mean ...
0
votes
1answer
200 views

When Leavitt path algebras are unital

I’ve been taking a seminar course on Leavitt path algebras, and our source material is rather laconic as far as explanations are concerned. I am attempting to justify (to myself) the following claim ...
3
votes
1answer
97 views

Finitely generated idempotent ideal

Let $R$ be a ring with identity. If $I$ is a finitely generated ideal of R such that $I^2=I$, must $I$ be generated by an idempotent? I've known that it holds for $R$ commutative. So I'm ...
0
votes
3answers
63 views

Zero divisors in matrix rings [closed]

Let $R$ be a commutative ring, $P \in M_n(R)$ and $\det(P)$ is a zero divisor of $R$. Must $P$ be a zero divisor of $M_n(R)$? Here rings mean unital rings, $M_n(R)$ denotes the ring of square ...
2
votes
1answer
52 views

Powers in non-commutative rings

Let $a,b$ be elements of a non-commutative ring $R$ with $\operatorname{char}(R) =p > 0$ and suppose that $ab-ba=[a,b]=1$. My question is simply: Could you give a formula for the element $(a^n ...
0
votes
0answers
27 views

Embedding a ring in a direct product

If an $R$-module $C$ is a homomorphic image of a direct sum $⊕M$, where $M$ is an $R$-module, and $R$ could be embedded in a direct product $ΠC$, could $R$ be embedded in a direct product $ΠM$?
5
votes
2answers
364 views

Is a finitely generated projective module a direct summand of a *finitely generated* free module?

Let $R$ be a (not necessarily commutative) ring and $P$ a finitely generated projective $R$-module. Then there is an $R$-module $N$ such that $P \oplus N$ is free. Can $N$ always be chosen such that ...
1
vote
0answers
7 views

Construction of the decomposition of semisimple algebra

I'm a theoretical phycysist and I'm working on the theory of quantum information, in which some problems are connected with properties of matrix algebras. My question is: If we know, that given ...
3
votes
1answer
60 views

Coker of powers of an F in End(M)

If Coker of an $F$ in End$(M)$ is of finite length, where $M$ is Noetherian, is Coker and Ker of all powers of $F$ of finite length? Is the condition of being noetherian necessary?
0
votes
0answers
32 views

Projectivity of a certain module

Let $M$ be a generator for the category of left $R$-modules, and let we have an $R$-epimorphism $h$ from $R^{(X)}$ to an $R$-module $P$ which is projective relative to $M^{(X)}$ ($X$ is a set). I want ...
4
votes
1answer
576 views

Artin-Wedderburn decomposition of a particular group ring

I am trying to do a question from an algebra qualifying exam: Decompose the group ring $\mathbb{F}_5[S_3]$ as a product of simple rings. By Maschke's theorem since $\mathrm{char}(\mathbb{F}_5) ...
2
votes
1answer
50 views

Rank as norm on matrix

Could we consider matrix rank $r$ a norm? Is other norm similar to rank $r$ possible to associate with a finite matrix? (We denote trivial valuation $|\alpha|=1,\quad\forall\alpha\in\Bbb F^*$ where ...
3
votes
1answer
43 views

Semisimplicity of a ring

As it is well-known, a ring with unity $R$ is semisimple if and only if each left $R$-module is projective. My question: Is simisimplicity of $R$ equivalent to each "simple" left $R$-module being ...
1
vote
1answer
21 views

Socle of a ring $R$

It is well-known that for an idempotent $e\in R$, the right $R$-module $eR$ is simple faithful if and only if $Re$ is a simple faithful left $R$-module. Now, I want to prove that when $Re$ is ...
0
votes
1answer
43 views

Semisimple modules and short exact sequences

Though I suspect that this is a folklore result, I've been unable to find a proof by digital book searching. I also mention (my apologies) that I am new to noncommutative algebras theory and I am not ...
1
vote
2answers
51 views

Example of the semisimple ring $R$ but $R^{{\rm op}}$ is not.

Is there any example of this kind of rings? i don't have any imagine of this rings, if they are exist!