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

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1answer
26 views

A simple divisible module

Let $k$ be division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\text{End}_k(V)$ be the ring of right linear transformations on $V$. My questions are: (1) Is $V$ a simple ...
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1answer
18 views

Using Exchange Lemma in a 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
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1answer
303 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 ...
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0answers
18 views

Simultaneous triangular form for nil algebras over division ring

Let $D$ be a division ring. Let $V$ be a finite dimensional module over $D$, let $I \subseteq\operatorname{End}_D(V)$ be a $D$-submodule on both sides (I mean a subgroup closed by both on left and ...
3
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0answers
42 views

What properties do the rings of infinite, upper-triangular matrices have?

I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field. But ...
8
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2answers
133 views

One-sided version of the Nakayama lemma?

The Nakayama lemma is often used to show that finitely generated idempotent ideals are generated by an idempotent. What remains true if we go to non-commutative rings? In other words, given a unital ...
0
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1answer
45 views

Annihilator of maximal ideals in a finite dimensional algebra

I wonder if the following is correct: The left (right) annihilator of every (2 sided) maximal ideal in a finite dimensional $k$-algebra is always nonzero. Clearly this is true for semi-simple ...
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1answer
24 views

Detailed example of a skew field different from Hamilton quaternion

Do you have a reference of a detailed construction of a skew field different from the quaternions from Hamilton? I would appreciate if that would be accessible from the Internet.
2
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1answer
37 views

If $R$ is a simple ring, is every corner $eRe$ simple?

Assume that $R$ is a ring, not necessarily commutative nor unital. Let $e$ be an idempotent in $R$. Is there a correspondence between ideals of $R$ and ideals of the corner ring $eRe = \{ere : r\in ...
3
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1answer
20 views

Question about Jacobson's proof of structure theorem for semi-simple Artinian rings

My question pertains to the proof of Proposition 4.7 on page 203 of Jacobson's Basic Algebra II (the Dover edition). The proposition says the following: if $R$ is semi-simple (which for Jacobson means ...
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0answers
16 views

Survey, text for noncommutative Grobner basis.

This is a survey/ text request for noncommutative grobner basis. A googling gave me these: http://www.sciencedirect.com/science/article/pii/0304397594902836 ...
3
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1answer
58 views

Literature on noncommutative rings

I am looking for books or notes about non commutative rings with with a maximum of data exposed without the help of modules (because I have many references which deal with the subject but modules are ...
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0answers
41 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
2
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1answer
40 views

When is the tensor product commutative?

I am working with the the tensor product $-\otimes_R -$ over some noncommutative ring $R$. Is the tensor product always commutative if $R$ is commutative? If so, can the tensor product be commutative ...
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1answer
414 views

Hom-tensor adjunctions

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 ...
2
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1answer
68 views

When is $M\otimes N$ a module?

So suppose we have a $R$-$S$-bimodule $M$, and an $S$-$T$ bimodule $N$, then we can construct the abelian group $M\otimes_S N$. Under what conditions could we make this a module? I would expect this ...
5
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1answer
145 views

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 ...
2
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1answer
48 views

Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
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3answers
68 views

Noncommutative algebraic operation. [closed]

Can we always find a non-commutative algebraic operation in a non-empty set?
2
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2answers
56 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 $R={\prod}_{i\in I} R_i$?
2
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4answers
219 views

Global dimension of free algebra.

Is there any easy way to see the global dimension of a free algebra $$ A=k\langle x_{1},\dots,x_{n} \rangle $$ is 1?
3
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1answer
44 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.
0
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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 ...
7
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0answers
149 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 ...
4
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1answer
440 views

Trivial extension of an algebra

Suppose that $A$ is a finite dimensional $k$-algebra. Call $Q=\mathrm{Hom}_k(A,k)$. $Q$ admits an $A$-$A$-bimodule structure in the obvious way. The trivial extension of $A$ is defined as follows: ...
4
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1answer
65 views

Semisimplicity is equivalent to each simple left module is projective?

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 semisimplicity of $R$ equivalent to each simple left $R$-module being ...
4
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2answers
264 views

Is the center of a ring an ideal?

Let $Z(R) = \{ a \in R : ax = xa,\text{ for all $x \in R$}\}$ Is $Z(R)$ an ideal of $R$? Attempt: I already proved that $Z(R)$ is a subring of $R$. I would say yes, since if $x \in R$, then $xa$ is ...
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4answers
663 views

Coproduct in the category of (noncommutative) associative algebras

For the case of commutative algebras, I know that the coproduct is given by the tensor product, but how is the situation in the general case? (for associative, but not necessarily commutative algebras ...
0
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0answers
43 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 ...
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0answers
22 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 ...
8
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1answer
56 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 ...
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2answers
43 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 ...
3
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1answer
41 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
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2answers
71 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
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1answer
75 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 ...
7
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2answers
168 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
4
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1answer
154 views

Matrices $B$ that commute with every matrix commuting with $A$

There have been many questions in the vein of this one, but I can't find one that answers it specifically. Suppose $A,B\in M_n(\mathbb C)$ are two matrices such that, for any other matrix $C\in ...
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0answers
24 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 ...
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2answers
196 views

Reduced norms of matrix algebras

I'm trying to understand a few basic notions on the reduced norm of division algebras, and more specifically the relation between the norm of an algebra and the norm of algebras similar to it. ...
0
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0answers
21 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) := ...
0
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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 ...
3
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2answers
103 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 ...
3
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1answer
71 views

Coker of powers of an endomorphism

Let $F\in\operatorname{End}_R(M)$, where $M$ is a Noetherian $R$-module. If $\operatorname{Coker}F$ is of finite length, is Coker and Ker of all powers of $F$ of finite length? Is the condition of ...
2
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1answer
53 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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0answers
26 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
30 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
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2answers
69 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
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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: ...
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1answer
35 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
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1answer
32 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 ...