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Results tagged with ra.rings-and-algebras
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user 345
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
22
votes
Accepted
Are there any finitely generated artinian modules that are not Noetherian?
Suppose you have an Artinian but not Noetherian finitely generated $R$ module $M$. Let $0\leq M_1\leq M_2\leq \cdots \leq M_n=M$ be a finite chain of $R$-modules such that each composition factor $M_i …
- 3,545
7
votes
Faithful flatness and non-commutative algebras
If $A$ happens to be (left) Noetherian then to show $f$ makes $B$ a faithfully flat right $A$-module it is enough to check that $B\otimes_A S\neq 0$ for every simple (left) $A$-module $S$ since, in th …
- 3,545
6
votes
Maximal Ideals in Formal Laurent Series Rings?
I think that the question difficult as illustrated by Hailong's answer. I suspect that it will be hard to even find a nice parameterisation of the $H$-orbits of maximal ideals in your refined question …
- 3,545
5
votes
$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $\ell^1(G)$ is a flat (right) $\mathbb{Z}G$-mod...
Here are a few observations that are too long for a comment but don't provide a full answer by any means.
As Peter Samuelson observed left-right questions aren't significant in this setting since $\m …
- 3,545
5
votes
Accepted
Testing ideal membership in the Weyl algebra: a simple example
Following my nose gave the following argument. Writing $I$ be the left ideal generated by $x\partial^2$ and $x^3$ and using $\cdot$ to stress multiplication we get
$$ x^2 \cdot x\partial^2 - \partia …
- 3,545
4
votes
Accepted
$R/I\cong R/\text{Ann}_R(R/I)$ but $I\neq\text{Ann}_R(R/I)$
You might as well assume that $\mathrm{Ann}_R(R/I)=0$ since if $S=R/\mathrm{Ann}_R(R/I)$ then $R/I\cong S$ as $R$-modules if and only if $S/(I/\mathrm{Ann}_R(R/I))\cong S$ as $S$-modules.
So now the q …
- 3,545
4
votes
Accepted
Dual of a bimodule
Copied from comments as requested.
There isn't enough context in the question but if you know that $B$ is a projective left $R$-module and nothing else there is a fair chance that you want $\mathrm{H …
- 3,545
4
votes
Accepted
Motivation and reference for Brauer algebras
For motivation I would advise starting with Brauer's original paper. You'll need a JSTOR login though:
https://www.jstor.org/stable/1968843?origin=crossref&seq=1#metadata_info_tab_contents
- 3,545
3
votes
Accepted
Is the pair $(C([0 \;1]),\mathbb{C})$ a consecutive pair?
The answers to the question prime ideals in C([0,1]) explain why the answer is no, assuming that Yemon Choi was mistaken in his answer in believing that you are only interesting in continuous homomorp …
- 3,545
3
votes
Accepted
Separable and finitely generated projective but not Frobenius?
Theorem 4.2 of On separable algebras over a commutative ring says that the answer is always yes.
- 3,545
2
votes
Comparing lower central series and augmentation ideal completions
I don't quite follow your definition of the mod $p$ lower central series as $s$ only seems to appear once in the definition. However whatever it is the answer is no.
If $G=\mathbb{Z}$ then the $I$-ad …
- 3,545
1
vote
1
answer
755
views
What are in units of an affinoid algebra?
Suppose that $K$ is a complete local field and $A$ is an affinoid $K$-algebra. Is there a known way to produce an explicit description of the units of $A$?
Here is what I already know: write $A^\circ …
- 3,545