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

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0
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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 ...
3
votes
0answers
19 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 ...
1
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0answers
34 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. ...
2
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0answers
53 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 ...
1
vote
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 ...
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) = ...
0
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1answer
337 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 ...
7
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2answers
319 views

Example of a commutative perfect ring that is not artinian

I read a result here stating that a commutative perfect ring is artinian if and only if it is $(1,1)$-coherent (see Proposition 5.3). I'm interested in finding an example of a commutative perfect ...
1
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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
votes
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 ...
0
votes
0answers
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 ...
5
votes
1answer
54 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 ...
0
votes
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
votes
2answers
50 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 ...
3
votes
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. ...
0
votes
1answer
99 views

Ideals in direct product of algebras over a field [duplicate]

Let $ B_1,...,B_n$ be $k$-algebras ($k$ is a field), $ B=\prod_{i=1}^{n}B_i $ their direct product, and $ J_i$ an ideal of $B_i$. I must to prove that the direct product $ J=\prod_{i=1}^{n}J_i $ ...
2
votes
2answers
187 views

Ideal of nilpotent elements in non-commutative ring.

Let $R$ be a non-commutative ring such that every element is either invertible or nilpotent. I am trying to show that the set of nilpotent elements, denoted $I$, is a two sided ideal, but I am ...
2
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0answers
39 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
votes
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 ...
1
vote
1answer
434 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 ...
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
votes
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 ...
0
votes
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 $i$, ...
2
votes
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
votes
0answers
51 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
votes
2answers
138 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
votes
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 ...
1
vote
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
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
votes
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
18 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
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 ...
5
votes
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 ...
2
votes
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 ...
3
votes
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 ...
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 ...
2
votes
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 ...
-2
votes
3answers
68 views

Noncommutative algebraic operation. [closed]

Can we always find a non-commutative algebraic operation in a non-empty set?
2
votes
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 $R={\prod}_{i\in I} R_i$?
2
votes
4answers
224 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
votes
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.
0
votes
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 ...
7
votes
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 ...
4
votes
1answer
456 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
votes
1answer
69 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
votes
2answers
278 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 ...
10
votes
4answers
685 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
votes
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 ...
8
votes
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 ...