All Questions
11 questions
3
votes
2
answers
468
views
How fast does the number of "fixed" points grow compared to the size of the ball in the following group?
I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb ...
- 823
11
votes
1
answer
159
views
Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?
A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that
$$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$
An example of such an object is ...
- 301
28
votes
2
answers
863
views
$A^2$ is isomorphic to $A^{(\omega)}$, but not $A$
Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)?
Edited to add: As no answers are forthcoming, does anyone ...
- 18.7k
0
votes
0
answers
150
views
How to classify rings by combinatorial structures?
There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
5
votes
1
answer
132
views
Integral monoid rings and Ore conditions
Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$.
I have two, ...
- 2,452
2
votes
1
answer
160
views
MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra
Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$.
Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-...
- 1,755
2
votes
1
answer
310
views
Flatness of submodules of free modules
Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group.
If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $...
- 5,901
5
votes
0
answers
187
views
Any f.p. faithful simple module over a primitive group ring?
Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$.
There ...
- 1,528
14
votes
1
answer
581
views
Group rings such that every (countably generated) module has a maximal submodule
Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma.
I am interested in the following question, with two variants.
...
- 38.6k
1
vote
1
answer
271
views
Rank of a locally free $\mathbb Z[G]$-module
This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...
- 255
3
votes
2
answers
2k
views
Extension problem
As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
- 2,857