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

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27 views

What do we call the function $R \rightarrow \mathrm{End}_{\mathbf{Set}}(X)$ associated with an $R$-module $X$?

First, a convention: given a mathematical structure $X$, write $\mathrm{End}_{\mathbf{Set}}(X)$ for the set of all functions $$X \rightarrow X.$$ Now let $R$ denote a ring. Question. Given an ...
3
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0answers
28 views

Injective hull of a simple module

Let $M$ be an indecomposable injective right module over a right Artinian ring $R$, so $M$ has exactly one associated prime ideal $P$ (Lectures on Modules and Rings, T.Y. Lam). Now, $R/P$ is a simple ...
2
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0answers
61 views
+50

What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $S \leftarrow ...
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1answer
11 views

Associativity of ore-polynomials

I'm fairly new to ore-polynomials. Until now I thought the rule $\frac{d}{dt}x(t)=\dot{x}(t)+x(t)\frac{d}{dt}$ resp. $x(t)\frac{d}{dt}=\frac{d}{dt}x(t)-\dot{x}(t)$ would do the trick in right- or ...
0
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0answers
16 views

If $A$ is a noncommutative ring then all biderivation is inner.

Before the question I will post some definitions: Derivation: An additive map $\delta: A \longrightarrow M$, where $A$ is a ring and $M$ is a $(A,A)-$bimodule is called derivation if $\delta(xy) = ...
2
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0answers
42 views
+50

A Problem for Nil-Ideals

Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for non-commutative rings I think this may not be true. ...
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0answers
30 views

Bochner-style theorem for SO(3)

Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar ...
0
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0answers
23 views

How many degree m elements in the exterior algebra on n generators over a finite field, vanish when raised to the r-th power?

Let $R=\Lambda_{\mathbb{F}_p}[e_1,...,e_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements (this arises naturally as the mod-p cohomology ring of the $n$-dimensional ...
0
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0answers
7 views

Nakayama automorphism $\sigma$ of Hecke Algebra $^0H^f_n$ is not inner for $n\geq 3$?

With $R=\mathbb{Z}[q_1,q_2]$, the Hecke algebra $H^f_n$ of $S_n$ is defined to be the $R$-algebra generated by $T_1,\dots,T_{n-1}$ satisfying $T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1}$, $T_iT_j=T_jT_i$ if ...
1
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1answer
59 views

If $[A,A]A[\lambda,A] = 0$ then $\lambda \in Z(A).$

Suppose that $A$ is a unital ring and $([A,A]) = A.$ If $[A,A]A[\lambda,A] = 0$ prove that $\lambda \in Z(A).$ Comments: This is part of an exercise I'm doing, I'm posting this part because I am ...
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31 views

Prove that $E(x)y + E(y)x \in Z(A)$ for all $x,y \in A$.

Let $A = M_2(C)$, where $C$ is any commutative ring. Define $E: A \longrightarrow A$ by $E(x) = x - tr(x)Id_2$, where $tr(x)$ denotes the trace of $x$. Prove that $E(x)y + E(y)x \in Z(A)$ for all $x,y ...
0
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1answer
38 views

Show that $([A,A])$ contains the identity matrix.

Suppose $A = M_n(C)$, $n \geq 1$, where $C$ is a commutative unital ring. If $n \geq 2$, then $([A,A])$, the ideal generated by $[A,A]$, contains the identity matrix.
3
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1answer
46 views

Adjoining an identity to a ring

I am run into the following in an Algebra text: "Let $R_0=\mathbb Z/2\mathbb Z⊕\mathbb Z/2\mathbb Z⊕\cdots$ viewed as a ring without identity, with addition and multiplication defined componentwise. ...
3
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2answers
53 views

Can we construct a homomorphism from a projective module into a free module?

In short, I have a projective module and a free module, and want to construct a module homomorphism between the two. Is this always possible, at least in some way? Let me go into more detail. Suppose ...
2
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0answers
40 views

Homological Conjectures

Let The strong Nakayama conjecture : If $M \in \rm{{mod\mbox{-}}}R$ and $\rm{Ext}^i(M,R)=0$ for $i \geq 0$, then $M$ is zero. The generalized Nakayama conjecture If $S$ is a simple module and ...
0
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1answer
90 views

How do modern algebraists think about diagonal matrices?

Let $\mathbb{K}$ denote a field and $A$ denote a $\mathbb{K}$-algebra. Then given a $\mathbb{K}$-subalgebra $\Delta$ of $A$, I suppose it make sense to declare that $m \in A$ is ...
5
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1answer
55 views

What shall I learn in order to understand Auslander-Reiten theory and tilting theory?

I work on cluster algebras and quivers and hence I need to understand Auslander-Reiten theory and tilting theory as soon as possible. I have read some noncommutative algebra and homological algebra ...
1
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0answers
10 views

Can MAGMA write Groebner basis elements in terms of the original generators?

Consider the free algebra $F = \mathbb Q(a)\langle x, y, z\rangle$ and the ideal $$I = \langle xy - ayx, yz - zy, xz - zx - y\rangle$$ According to the following code $y^2 \in I$. ...
2
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1answer
38 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 ...
2
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0answers
23 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
52 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 ...
0
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1answer
46 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 ...
3
votes
1answer
41 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
22 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 ...
0
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0answers
19 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 ...
5
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0answers
42 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 ...
3
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1answer
63 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 ...
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
28 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.
3
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1answer
74 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 ...
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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
58 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
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1answer
46 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.
5
votes
1answer
151 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 ...
0
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1answer
12 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
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0answers
46 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 ...
3
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1answer
43 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
45 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
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2answers
88 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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0answers
152 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
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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 ...
0
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0answers
23 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
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1answer
59 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
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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
25 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
78 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
104 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
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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 $e \in I$.
2
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1answer
56 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
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1answer
31 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: ...