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

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-1
votes
3answers
57 views

Noncommutative algebraic operation. [on hold]

Can we always find a non-commutative algebraic operation in a non-empty set?
3
votes
2answers
50 views

Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
3
votes
1answer
41 views

A ring isomorphic to its square [duplicate]

Is there an example of a ring $A$ (with unity) which is isomorphic as unital rings to $A\times A$? Any such ring can't have invariant basis number so in particular can't be commutative.
3
votes
1answer
93 views
+50

What is $\operatorname{Hom}_R(P,R)$ isomorphic to when $P$ is projective?

Let $R$ be a (possibly noncommutative) ring with $1$. Now, quite clearly we have $$\operatorname{Hom}_R(R^n,R)\cong R^n.$$ I am wondering if there is any similar result for ...
0
votes
1answer
11 views

What conditions make the ring of Laurent polynomials in non-commuting variables countable?

Suppose we have some commutative ring $R$ and the ring of Laurent polynomials in a finite number of non-commuting variables $S=R\langle x_1,\,x_1^{-1},\ldots,\,x_n,\,x_n^{-1}\rangle$. Under what ...
0
votes
0answers
38 views

How bad must be a ring to allow cyclic artinian modules that are not noetherian?

I've been studying the relations between artinian and noetherian modules over commutative rings. One can prove two interesting results for the commutative case. Theorem Every commutative artinian ...
1
vote
0answers
18 views

Dual of Osofsky Theorem

A theorem of Osofsky reads: "A ring $R$ is semisimple iff the intersection of two injective submodules of any right $R$-module is injective" (Exercises in Modules and Rings, T.Y.Lam, Ex.3.11). Does a ...
3
votes
1answer
37 views

Endomorphism ring of $x \Bbbk\langle x,y \rangle + y \Bbbk\langle x,y \rangle$

Let $R = \Bbbk \langle x,y \rangle$, where $\Bbbk$ is a field. I want to determine $\underline{\text{End}}_R(xR + yR)$, the ring of (not necessarily degree-preserving) graded module homomorphisms ...
0
votes
2answers
40 views

Flat modules and their relationship with short exact sequences

I recently came across the following result on a Wikipedia page: Suppose $0\to A\to B\to C\to 0$ is a short exact sequence where $B,\,C$ are flat modules; then $A$ is a flat module. I wanted to ...
0
votes
2answers
66 views

Is it true that a flat module is torsion free over an arbitrary ring? Does the reverse implication hold for finitely generated modules?

So when you work over a commutative ring, this result is quite well known. I am wondering if the same holds true for an arbitrary ring; that is, if $R$ is some (possibly noncommutative) ring, does the ...
7
votes
0answers
146 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
0
votes
0answers
21 views

free algebras over noncommutative rings

For a commutative ring $R$ and a set $X$, we can regard the polynomial algebra $R[X]$ as the free commutative $R$-algebra on $X$. For a unital associative ring $R$ which is not necessarily ...
0
votes
0answers
14 views

relationship of semicircular and circular elements, free fock space

I am trying to understand an argument in the paper Limit laws for Random matrices and free products by Dan Voiculescu, p 212. Let $\mathscr{T}\left(H_{n}\right) := ...
8
votes
1answer
54 views

Two more questions on Kontsevich's “Noncommutative Identities” (Derivations on $\mathbb{C}\langle X,Y \rangle$) [Solved]

The following two questions regard once more the following article: arXiv:1109.2469. In the second chapter we are dealing with the Lie Algebra $\mathfrak{g}$ of derivations $\delta$ of ...
0
votes
1answer
14 views

Using Exchange Lemma in an decomposition

The "Exchange Lemma" in the decomposition of modules states that any decomposition of right $R$-modules $M_1⊕...⊕M_n=A⊕B$ with the endomorphism ring $End (A_R)$ local yields $M_i≈A⊕X$ for some $i$, ...
0
votes
1answer
24 views

What is a formal model for equations in non-commutative ring?

The standard formal modeling for polynomials is the polynomial ring $R[X_1,...,X_n]$ which is a monoid ring $R[\mathbb{N}^n]$ over an rng $R$. Under this construction, it is possible to commute ...
7
votes
1answer
73 views

Does $M_n(R_1)\cong M_n(R_2)$ imply $R_1\cong R_2$?

Let $R_1,R_2$ be two rings with identity. If for some $n\in\mathbb N$, $M_n(R_1)$ and $M_n(R_2)$ are isomorphic as rings, can we deduce that $R_1\cong R_2$? I can prove it when both $R_1,R_2$ are ...
3
votes
2answers
102 views

Algebras $A_i$ generated by a single element over an infinite field, does $A_1 \times \cdots \times A_r$ has the same property?

Let $K$ be an infinite field and $A_1, \ldots, A_r$ finite dimensional algebras over $K$ and such that $\forall i = 1, \ldots, r \ \exists x_i \in A_i : A_i = K[x_i]$. (I think we say that $A$ is ...
1
vote
0answers
24 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 $e \in I$.
2
votes
1answer
52 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 ...
0
votes
1answer
29 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
54 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
34 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
25 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
17 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 ...
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 $V$ ...
2
votes
2answers
26 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
54 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
63 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 $\mathbb{Z}$, ...
4
votes
1answer
107 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
1answer
32 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
33 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) ...
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 ...
1
vote
0answers
25 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 ...
1
vote
1answer
44 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 ...
5
votes
0answers
77 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 ...
0
votes
0answers
51 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 ...
1
vote
1answer
33 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 ...
1
vote
0answers
54 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
votes
1answer
38 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
39 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. ...
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 ...
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 ...
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
37 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
45 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 ...