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

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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 $e \in I$.
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1answer
41 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 ...
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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: ...
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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 ...
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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: ...
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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 ...
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1answer
22 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 ...
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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 ...
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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 ...
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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 $V$ ...
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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 ...
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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 ...
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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 ...
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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 $\mathbb{Z}$, ...
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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 ...
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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 ...
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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 $p \in A$ ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
28 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
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 ...
2
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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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29 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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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 ...
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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 ...
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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 ...
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32 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
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 ...
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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 ...
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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 ...
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3answers
62 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
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 ...
2
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1answer
51 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
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 $ΠM$?
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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 ...
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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 ...
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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 ...
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1answer
20 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
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!
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1answer
52 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
50 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 ...
2
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1answer
96 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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25 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
48 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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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 ...