Tagged Questions
2
votes
1answer
54 views
Proving part of the Wedderburn Structure Theorem
I'm having trouble with an exercise from "Noncommutative Algebra," by Farb & Dennis proving part of the Wedderburn structure theorem for semisimple rings.
If $R$ is a semisimple ring and ...
4
votes
1answer
29 views
Rings with finitely many finitely generated free modules, up to isomorphism
If $A = \mathrm{End}(V)$, where $V$ is an infinite-dimensional vector space over some field, then it's not hard to see that $A \cong A^2 \cong \dotsb$. In particular, the map $\mathbb{Z} \to K_0(A)$ ...
5
votes
3answers
66 views
Commutative property of ring addition
I have a simple question answer to which would help me more deeply understand the concept of (non)commutative structures. Let's take for example (our teacher's definition of) a ring:
Let $R\neq ...
3
votes
0answers
57 views
When is $\mathbb{Z}\Gamma$ a left Noetherian ring?
Denote $\Gamma$ to be a countable discrete group, let $\mathbb{Z}\Gamma$ to be its integer group ring.
A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals.(c.f. ...
4
votes
1answer
96 views
Center of the unit group $R^\times$ of a ring.
Let $R$ be a (unital, noncommutative) ring.
If $R^\times$ is the unit group of $R$ (the set of elements with a two-sided inverse), it is clear that $R^\times \cap Z(R) \subset Z(R^\times)$, where ...
5
votes
1answer
72 views
Noncommutative Hilbert basis theorem is false?
How can I show that for a field $K$, in the free algebra on $2$ generators $K\langle x,y\rangle$, the two-sided ideal $$\big\langle\!\big\langle xy^ix\;\big|\;i\in\mathbb{N}\big\rangle\!\big\rangle ...
2
votes
0answers
41 views
Left ideals of central simple algebra generated by symmetric element
Let $F$ be a field of characteristic $\neq 2$. Suppose $(A,\sigma)$ is central simple algebra over a field $F$ with an orthogonal involution. Assume $(A,\sigma)$ is non-split. How to show that every ...
3
votes
1answer
50 views
socles of semiperfect rings
For readers' benefit, a few definitions for a ring $R$.
The left (right) socle of $R$ is the sum of all minimal left (right) ideals of $R$. It may happen that it is zero if no minimals exist.
A ring ...
0
votes
1answer
40 views
How to prove finite dimensionality of Hom-spaces between modules of finite length?
Let $R$ be a ring and $k$ be a field such that $k\hookrightarrow R$. Thus, given two $R$-modules $M$ and $N$, we can regard $\operatorname{Hom}(M,N)=\operatorname{Hom}_R(M,N)$ as a vector space over ...
3
votes
1answer
68 views
Natural homomorphism $\mathfrak{a} \prod E_\lambda \to \prod \mathfrak{a} E_\lambda$
Let $A$ be a ring (we don't assume that $A$ is commutative), and $\frak a$ a left ideal of $A$. Let $(E_\lambda)$ be a system of left $A$-modules. There is a natural homomorphism of modules
$$ \phi ...
4
votes
1answer
47 views
What is the definition of 'regular local' and 'regular' for noncommutative rings?
I have been trying to find out what the definition of a noncommutative regular local ring is, but to no avail. In fact, how does one even begin to define Krull dimension for a noncommutative ring? ...
2
votes
0answers
78 views
On the Nakayama functor
Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
3
votes
1answer
88 views
Splitting idempotents
Let $C$ be an additive category. An idempotent $e=e^2\in\mathrm{Hom}_C(X,X)$ is split if there are morphisms $\mu:Y\rightarrow X$, $\rho:X\rightarrow Y$ such that $\mu\rho=id_Y$ and $\rho\mu=e$. ...
3
votes
1answer
96 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 structire in the obvious way. The trivial extension of $A$ is defined as follows:
...
3
votes
1answer
70 views
Selfinjectivity and Frobenius algebras
Suppose that $R$ is a finite dimensional $k$-algebra. I say that $R$ is Frobenius if it is locally bounded (see this question for a definition) and indecomposable projectives and injectives coincide. ...
2
votes
1answer
48 views
On locally bounded algebras
Let $k$ be a field and $R$ an associative $k$-algebra suppose $R^2=R$. We say that $R$ is locally bounded if there exists a complete set of pairwise orthogonal primitive idempotents $\{e_x:x\in I\}$ ...
5
votes
1answer
128 views
Why are these all the indecomposable projective modules?
Let $A$ be a finite dimensional associative algebra with unit over a commutative field $k$. Suppose that $M$ is a finitely generated left module. Denote by $I(M)$ the injective hull of $M$ and by ...
0
votes
1answer
48 views
Trace map $Ext^i(E,E)\rightarrow H^i(X,O_X)$
Let $X$ be a scheme (or complex manifold if you like) and $E$ be a sheaf on $X$. I would like to know the definition of so-called trace map
$$Ext^i(E,E)\rightarrow H^i(X,O_X)$$
for $1\le i\le \dim X$.
...
2
votes
0answers
69 views
Non-commutative integral extensions?
In Commutative algebra there is a notion of an integral extension: Let $P$ be a subring of $R$. Then $R$ is the integral extension of $P$ if each element of $R$ is a root of a monic polynomial with ...
3
votes
2answers
137 views
Intersection of a subring and an ideal
Given a unital ring $R$ and its unital subring $P$ (with the same unit). Also, given a maximal left ideal $L$ of $R$. Must $L\cap P$ be a maximal left ideal of $P$? I don't think so, but are there any ...
3
votes
0answers
61 views
Applications of Govorov-Lazard Theorem?
The Govorov-Lazard Theorem states that a (right) module over an unital ring is flat iff it is a direct limit of finitely generated free (right) modules.
I wonder if this theorem has interesting ...
1
vote
1answer
85 views
Commutants of commutative algebras
Let $W$ be a unital algebra and let $V$ be its maximal abelian subalgebra. Must the commutant $V^\prime$ of $V$ be commutative?
3
votes
1answer
168 views
Intersection of principal ideals
An intersection of principal left ideals need not be principal but incidentally this phenomenon is witnessed in von Neumann regular rings. How about arbitrary intersections of infinitely many ...
2
votes
2answers
96 views
Noncommutative rings, finding $a$ and $b$ such that every term in the sum $(a+b)^n = a^n + a^{n-1}b + \ldots$ is distinct
This question was inspired by the binomial theorem for rings. For commutative rings, we have the identity
$$(a+b)^n = \sum_{k=0}^n {n \choose k}a^kb^{n-k}$$
which does not hold for non-commutative ...
8
votes
1answer
149 views
Tensor product of simple modules
Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. My questions are:
How can we describe $M \otimes_R N$ explicitly? Well, I guess that it is a quotient of $R$ by a sum of ...
2
votes
1answer
139 views
Relaxing the definition of a von Neumann regular ring
Hereinafter, all rings are assumed to be unital but not necessarily commutative. A well-known class of rings are von Neumann regular rings, that is, rings $R$ such that for each $a\in R$ there is an ...
1
vote
3answers
119 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?
1
vote
0answers
108 views
All finite-dimensional simple modules are $1$-dimensional
Let $A$ be a (non-commutative) $k$-algebra, where $k$ is an algebraically closed, characteristic zero field. Let $M$ be a finite-dimensional simple $A$-module. If $A/\operatorname{ann}(M)$ is ...
2
votes
1answer
89 views
Localization of a non-commutative ring
Let $A$ be the non-commutative ring given by
$$
A=\mathbb{C}\langle x,y,z \rangle /(xy=ayx,yz=bzy,zx=cxz)
$$
for some $a,b,c\in \mathbb{C}$. What is the localization $A_{(x)}$ of A with respect to the ...
10
votes
1answer
171 views
Does this notion of morphism of noncommutative rings appear in the ring theory literature?
Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on ...
2
votes
1answer
67 views
Diamonds of ideals, part 3
I'd like to wrap up the line of questioning started first in this question and then continued in this question.
The only variant left to try is:
"How close can you get to the Diamond lattice ...
2
votes
1answer
129 views
Is every semi-simple ring a product of simple rings
I was wondering if the following statements were true;
1) Every semi-simple ring is a product of simple rings.
2) Every module over a division ring $R$ is free.
I think both of these statements are ...
4
votes
1answer
144 views
invertible polynomials over non-commutative rings
Let $f = a_0 + a_1 t + \dotsc + a_n t^n$ be a polynomial over some nontrivial, possibly noncommutative ring $R$. When is $f$ invertible in $R[t]$?
When $R$ is commutative, the answer is well-known: ...
5
votes
0answers
111 views
Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?
It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
1
vote
1answer
167 views
Finding all simple $R$ modules of a ring.
I was hoping someone had an idea on how to go about solving the following;
Find (up to isomorphism) all simple R-modules where
i) $R =
\begin{pmatrix} \mathbb{Z}/15 \mathbb{Z} & \mathbb{Z}/15 ...
2
votes
1answer
95 views
On the length of a ring
Suppose that $R$ is a ring, and suppose that $\lambda_R(_RR)<\infty$ and $\lambda_R(R_R)<\infty$ (where $\lambda_R$ is the length of an $R$-module). Is it true that then ...
2
votes
1answer
208 views
The Jacobson Radical of a Matrix Algebra
I am trying to solve the following question.
Let $A$ be the algebra over $\mathbb{C}$ consisting of matrices of the form
\begin{pmatrix}
* & * & 0 & 0 \\
* & * & 0 & 0 \\
* ...
5
votes
3answers
301 views
The Jacobson radical and quasi-regular elements in polynomial rings — trouble understanding a proof.
The concept of the Jacobson radical is quite new to me and I have to give a talk about a certain paper concerning it. I agreed to do this to find motivation to finally study this concept properly, as ...
3
votes
0answers
142 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 ...
5
votes
3answers
143 views
Can the product of non-zero ideals in a unital ring be zero?
Let $R$ be a ring with unity and $0\neq I,J\lhd R.$ Can it be that $IJ=0?$
It is possible in rings without unity. Let $A$ be a nontrivial abelian group made a ring by defining a zero multiplication ...
3
votes
3answers
152 views
Why is this ring semisimple?
Let $R$ be a simple ring (i.e. a ring with no nontrivial two-sided ideals) which contains a left ideal which is simple as a left $R$-module. How can I prove that $R$ is semisimple?
2
votes
3answers
164 views
Is $AxA$ a two-sided ideal for an element $x$ of a ring $A$?
In Bourbaki's Algebra there is the following proposition:
Let $A$ be a ring (with $1$), $(x_\lambda)_{\lambda\in L}$ a family of elements of $A$ and $\mathfrak{a}$ the set of sums $\sum_{\lambda\in ...
-1
votes
2answers
162 views
Sided inverses in a non-commutative ring
I've asked myself the following question : does there exist a non-commutative ring $R$ with unity $1$ and elements $x,y,z \in R$ such that $xyz = 1$ but $y$ has no left nor right inverses?
(Perhaps I ...
1
vote
0answers
83 views
Unit group of quotient of noncommutative polynomial ring
In this recent post the original question led people to look for rigid, noncommutative rings. (Rigid means that the only endomorphisms are zero and the identity). Several (somewhat complicated) ...
10
votes
1answer
204 views
What are the rings in which left and right zero divisors coincide called?
A unital ring $R$ is reversible iff $ab=0\implies ba=0.$ This condition implies the following one.
If $a\in R$ is a left-zero divisor, then $a$ is also a right-zero divisor. And the other way ...
1
vote
0answers
166 views
A classic example in noncommutative ring theory
A classic example of a left-not-right Noetherian ring is the quotent of the free algebra $\mathbb{Z}\langle x,y\rangle$ (noncommuting indeterminates) by the ideal generated by $y^2$ and $yx$. (It's ...
37
votes
6answers
1k views
Is there a non-commutative ring with a trivial automorphism group?
This question is related to this one. In that question, it is stated that nilpotent elements of a non-commutative ring with no non-trivial ring automorphisms form an ideal. Ted asks in the comment for ...
6
votes
1answer
262 views
A graded ring $R$ is graded-local iff $R_0$ is a local ring?
Update: I've copied this question over to mathoverflow.net:
http://mathoverflow.net/questions/100755/a-graded-ring-r-is-graded-local-iff-r-0-is-a-local-ring
to see if I get any answers there.
Let ...
5
votes
0answers
155 views
A ring that has exactly 7 left ideals (T. Y. Lam)
Exercise 3.25 in Lam's First Course states:
Let $R$ be a ring that has exactly seven nonzero left ideals. Prove that one of them is an ideal (i.e. left and right) and provide an example of such a ...
7
votes
1answer
135 views
Proper ideals generated by central ideals
Let $R$ be a unital ring and denote its center by $Z(R)$. If $I$ is an ideal of $Z(R)$, then the set $RI$ (consisting of finite sums of elements of the form ra where $r\in R$ and $a\in I$) is clearly ...
