Questions tagged [group-rings]

A group ring $R[G]$ is a ring constructed from a group $G$ and ring $R$. A special case of this construction is group algebra, which occurs naturally in representation theory.

318 questions
Filter by
Sorted by
Tagged with
36 views

38 views

$Z\pi \cong Z\oplus Z$ and so $P \cong Z$ [closed]

If $\pi = Z$, then the augmentation ideal $P$ is projective and $0 \to P \to Z\pi \to Z \to 0$ is a projective resolution. Here we know that $P$ has basis $\mathbb{Z}-$ ${0}$ Then how to show that it ...
87 views

12 views

Is problem of finiteness in the sense of von Neumann field FG algebras solved for any class of groups?

there is an open problem: Does for every field $F$ of characteristic prime and every group $G$, the group ring $FG$ finite in the sens of von Neumann? Is this problem solved for any class of groups? I ...
60 views

Element of group ring not invertible, but invertible in bigger group ring.

I have thought about such problems: Is there a field $F$ and a group $G$ such that there exists element $x$ of group ring $FG$ which is not invertible in $FG$, but is invertible in $KG$, where $K$ is ...
55 views

Scalar extension of group algebras

Let $R$ be a commutative unital ring, let $G$ be a group and let $R[G]$ its group $R$-algebra. It is true that $\mathbb{Z}[G]\otimes_{\mathbb{Z}}R\simeq R[G]$? If not, there are conditions on $R$ and ...
50 views

How to show $\operatorname{Ext}^1(I, \mathbb{Z})\cong\mathbb{Z}/2$ for $\mathbb{Z}[C_2]$?

I am trying to become better acquainted with $Ext^1$ and thought the best place to start is with actual computations. This was an exercise in some brief notes I found and wanted to see if my thinking ...
194 views

The category of finite dimensional right $KG$-modules is equivalent to the category of finite dimensional representations of a quiver $Q$

Let $K$ be an algebraically closed field and $G$ be a finite group such that $|G|$ is not divisible by the characteristic of $K$ (so that Maschke's theorem can be applied). Let $Q$ be the quiver ...
35 views

Does every finite dimensional algebra contains a semisimple subalgebra?

As in title - does every finite dimensional algebra over field K contains a semisimple (or semiprime, as in finite dimension it is the same) subalgebra? Due to coronavirus I dont have my notes from ...
58 views

Structure of ideals in $\mathbb{F}_q[G]$

Let $\mathbb{F}$ be a finite field of characteristic $p$ and let $G$ be a cyclic group of order $p^n$. I read in a paper that all ideals of the group ring $\mathbb{F}[G]$ are of the form $I_n^j$ where ...
58 views

Dimension of center of k[G]/rad k[G] where characteristic of k divides the order of G.

Let G be a finite group and consider k[G] where k is a field. In the scenario where char(k) divides |G|, how can one show that the dimension of Z(k[G]/rad k[G]) is strictly less than dimension of Z(k[...