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

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11 views

Show that the set $\{[x_i,x_j]; i>j\}$ and $\{[x_i,x_j,x_k], i>j \leq k\}$ are linearly independent.

Show that the set $\{[x_i,x_j]; i>j\}$ and $\{[x_i,x_j,x_k], i>j \leq k\}$ are linearly independent. I need to prove it to show that it is based on certain sets, but I can not prove, I believe ...
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1answer
11 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 ...
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1answer
43 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 ...
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1answer
56 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 $\mathbb{Z}$, ...
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60 views
+50

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
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1answer
21 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 ...
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18 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 $p \in A$ ...
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0answers
30 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) ...
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0answers
36 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 ...
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21 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 ...
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1answer
34 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 ...
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72 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 ...
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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 ...
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1answer
25 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 ...
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0answers
49 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 ...
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1answer
27 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 ...
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0answers
22 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. ...
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0answers
50 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 ...
2
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1answer
29 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 ...
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1answer
12 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 ...
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0answers
30 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 $I$ ...
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0answers
32 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 ...
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2answers
34 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 ...
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0answers
17 views

Decomposition of kernel of an epimorphism

Let $f:P→M$ be a projective cover of a left $R$-module $M$ in the sense that $f$ is an $R$-epimorphism from a projective $R$-module $P$ to $M$ with a small kernel in $P$. Assume $P=P_1⊕P_2$ so that ...
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1answer
40 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 ...
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3answers
60 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 ...
3
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1answer
88 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 ...
2
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1answer
49 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 ...
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0answers
26 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 $ΠM$?
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6 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 ...
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1answer
41 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 ...
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1answer
40 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 ...
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1answer
14 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 ...
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2answers
50 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!
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1answer
49 views

How do I find the spectrum of a ring?

What is $Spec R$ where $R$ is the integers modulo $6$? More generally, what are the techniques to find the spectrum of any commutative ring? (I would also be interested in the non-commutative case but ...
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1answer
49 views

Prime which is not Irreducible in ring with unity without zero divisors (not necessarily commutative)

In a non-commutative ring with unity without zero divisors find a prime element which is not irreducible (if possible). $p$ is prime iff $p|ab$ implies that $p|a$ or $p|b$, and $x$ is irreducible iff ...
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1answer
94 views

Non-commutative ring (not necessarily with multiplicative identity) of order $n$ exists if and only if $p^2|n$ for some prime $p$?

Is it true that there is a non-commutative ring (not necessarily with unity) of order $n$ if and only if $p^2\mid n$ for some prime $p$ ?
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0answers
23 views

Tensor product and opposite rings

Suppose $A$ is a ring, $M$ is a right $A$-module, and $N$ is a left $A$-module. In this situation, we can form the tensor product, $M\otimes_A N$, and this is an abelian group (and even a ...
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1answer
46 views

Prime elements in a noncommutative ring

Is there a reasonable definition of prime element in a noncommutative ring? The definition from wikipedia makes the assumption of commutativity and I'd like to know how necessary this condition is. ...
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2answers
70 views

Showing minimal graded free resolutions are isomorphic

I'm currently reading Rogalski's notes on noncommutative projective algebraic geometry (which can be found here) and I'm currently trying to fill out the details of Lemma 1.24 (2). The step which I ...
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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 ...
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54 views

On the group algebra of a group specified by generators and relations

Let $k$ be a field, $G = \langle S \rangle/N(R)$ be a group specified by a set of generators $S$ and a set of relations $R$ (the brackets denote the free group, and $N(R)$ means the conjugate closure ...
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0answers
12 views

Generators Precisely Cogenerators

I want to prove that the following are equivalent: 1) a ring $R$ is simple artinian; 2) each nonzero left $R$-module is a generator; 3) each nonzero left $R$-module is a cogenerator. My try: if ...
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2answers
44 views

$K$-basis of cyclic $KG$-module determined by group elements

Let $G$ be a finite group and $K$ a field with $\mathrm{char}(K) \nmid |G|$, and let $M$ be a cyclic $KG$-module. For a cyclic generator $v \in M$ we can pick group elements $g_1, \dots, g_d \in G$ ...
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1answer
27 views

A question on decomposition of modules

Let $M$ be an $R$-module. A decomposition $M=⊕_A M_\alpha$ of nonzero submodules is said to complement direct summands in case for each direct summand $K$ of $M$ there is a subset $B$ of $A$ with ...
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1answer
40 views

A question about $rad(M)$

We know that for any $R$-module $M$ the radical $rad(M)$ is the sum of all small submodules of $M$. My question: "if $x\in rad(M)$ is it true in general that $Rx$ is a small submodule of $M$? ...
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1answer
30 views

Projective modules over enveloping algebra

How could I prove the following statement? Let $k$ a commutative ring and let $A$ an associative $k$-algebra that is also a projective $k$-module. Then every projective $A\otimes A^{op}$ left module ...
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1answer
10 views

Associative and anticommutative Binary Operation(composition)

Show that if binary operation ,$\Delta$, is associative and anticommutative on $\mathbb{E}$, then $x\Delta y \Delta z=x\Delta z$ ∀$x,y,z \in \mathbb{E}$. [Hint: consider $x\Delta y\Delta z\Delta ...
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0answers
39 views

Nilpotent Jacobson radical of $End(M)$

If $M$ is a Noetherian injective left $R$-module, is it true that the Jacobson radical $J=J(End(M))$ of the endomorphism ring of $M$ is nilpotent? I know that if a left $R$-module is Artinian or ...
5
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1answer
56 views

A free direct sum of a projective module

I want to prove that a left module $_RP$ is a projective generator if and only if a direct sum of (copies of) $P$ is free. My try is, first, to observe that $P$ is a generator if and only if for ...