All Questions
Tagged with noncommutative-algebra noetherian
20 questions
4
votes
1
answer
171
views
Are finitely generated subrings of $\mathbb{H}$ Noetherian?
Let $R$ be a finitely generated subring of the ring of real quaternions $\mathbb{H}$, that is, $R$ is the subring generated by a finite subset of $\mathbb{H}$. I want to show (or find a counterexample)...
user815463
2
votes
1
answer
129
views
(Left) Noetherian domains and Torsion submodules
By a domain I mean a non trivial ring without any zero-divisors (not necessarily commutative).
Let $R$ be a ring and $M$ be a left $R$-module. We say an element $m\in M$ is a torsion element iff ...
- 1,057
0
votes
1
answer
80
views
Noetherian modules and Noetherian rings
I want to show that if $R$ is a Noetherian ring then $Mat_n(R)$ is also a Noetherian ring. It is obvious that $Mat_n(R)$ is a finitely generated $R$-module. So $Mat_n(R)$ is a Noetherian R-module.
...
- 991
3
votes
0
answers
95
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Simplicity of Noetherian $B$, $A \subseteq B\subseteq C$, where $A$ and $C$ are simple Noetherian domains
After receiving important comments, which show that my original question has already been asked and answered, I now change my question to the following non-commutative setting:
Let $A \subseteq B \...
- 6,937
4
votes
1
answer
312
views
In a left noetherian ring, does having a left inverse for an element guarantee the existence of right inverse for that element? [closed]
In a left noetherian ring, does having a left inverse for an element guarantee the existence of right inverse for that element?
- 1,009
0
votes
0
answers
163
views
Intersection of all prime ideals of weakly Noetherian ring
Prove that intersection of all prime ideals is nilpotent in weakly Noetherian associative ring R.
Hello,
I've got stuck with this question. Could you please give any advice or reference to some ...
- 137
2
votes
0
answers
73
views
$R$ right Noetherian. Is it true that $R(x)\otimes_{R[x]}R(x)\cong R(x)$?
Let $R$ be a right Noetherian ring (actually it is left Noetherian as well) and $S=R[x]$ the polynomial ring in one (commuting) variable. If $X$ is the set of all monic polynomials then $X$ is a right ...
- 1,431
3
votes
1
answer
193
views
Is $R\langle x_1,\ldots,x_n\rangle$ Noetherian?
Let $R$ be a Noetherian integral domain of characteristic $0$, and let $R\langle x_1,\ldots,x_n\rangle$ be the free $R$-ring. Is $R\langle x_1,\ldots,x_n\rangle$ necessarily Noetherian?
1) The proof ...
- 31
0
votes
0
answers
308
views
The ring of quotients of the first Weyl algebra
Since there are no comments to this question, I now restrict it to the following question:
It is known that: A ring $R$ is a prime left Goldie ring if and only if $R$ has a left quotient ring which ...
- 6,937
1
vote
0
answers
117
views
A simple Artinian left quotient ring of a left Noetherian domain
Recall the following known result, Theorem 6.1: A ring $R$ is a prime left Goldie ring if and only if $R$ has a left quotient ring which is a matrix ring over a division ring (= simple Artinian).
Now,...
- 6,937
4
votes
1
answer
183
views
When does an integral group ring have finite global dimension?
Let $G$ be a finite group and $R=\mathbb{Z}[G]$ the integral group ring. If $G$ is such that $R$ is Noetherian (so $G$ polycyclic-by-finite) when does $R$ have finite global dimension? Another way of ...
- 257
4
votes
0
answers
164
views
When is $RS^{-1}$ a local ring?
Suppose we have a noncommutative ring $R$ and multiplicatively closed set that is both right Ore, and right reversible, i.e. it is a right denominator set. Now, we can localize $R$ at $S$ to form $RS^{...
- 1,431
5
votes
1
answer
286
views
Ore extensions of noetherian $k$-algebras are again noetherian (proof explanation)
Let $k$ be a field. All occuring $k$-algebras are required to be associative and unital. By noetherian I always mean left noetherian.
In a lecture I’m currently taking the notion of an Ore extension ...
- 16.8k
1
vote
1
answer
81
views
Uncountably many left ideals?
Let $R$ be a following subring of $M_2(\mathbb{C}):$
\begin{equation*}
R = \left\{ \begin{bmatrix}
a & r \\
0 & s
\end{bmatrix} ~:~ a\in \mathbb{Q} ~\mbox{...
- 544
2
votes
1
answer
1k
views
matrix ring Noetherian, Artinian, semisimple?
Let $k$ be a field and $\Lambda=\begin{bmatrix}
k & 0 \\
k^2 & k[x]/(x^2)
\end{bmatrix}$. This ring is an algebra over $k$.
(a) What is $\dim_k \Lambda$?
(b) Is $\Lambda$ a left ...
- 460
3
votes
1
answer
438
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 ...
- 135
2
votes
1
answer
283
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 ...
- 1,175
1
vote
1
answer
526
views
Noncommutative finitely generated algebras need not be noetherian
I would like to understand an example (of the title) given in the book "An Introduction to Noncommutative Noetherian Rings" by K. R. Goodearl, R. B. Warfield...
On page 8, Exercise 1E, an example of ...
- 521
1
vote
1
answer
303
views
Simple Noetherian domain which is not a division ring
I need a simple Noetherian domain which is not a division ring.
I do know that this ring must not be Artinian, since otherwise it would be a division ring. Thanks in advance!
- 5,848
3
votes
1
answer
144
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 ...
- 5,848