Articles of group rings

Conditions for free/projective/flat module over a groupring

Let $H \subset G$ be a subgroup of the group $G$. When is $\mathbb{Z}[G]$ a free/projective/flat $\mathbb{Z}[H]$-module? If $\mathbb{Z}[G]$ is a free $\mathbb{Z}[H]$-module then there is a $n \in \mathbb{N}$, such that $\mathbb{Z}[G] \cong \mathbb{Z}[H]^n$, as far as I know. Obviously if $G = H$ and $n=1$ this is true, but I don’t seem to […]

Jacobson radical of the integral group ring

I am trying to prove that the Jacobson radical of the integral group ring $\mathbb{Z}G$ for a finite group is zero. Most of what I find on semisimplicity, Jacobson semisimplicity, has to do with group algebras $KG$ where $K$ is a field. I did come across this, Jacobson radical of a ring finitely generated over […]

Is there a category theory notion of the image of an axiom or predicate under a functor?

Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I want to do (in particular, I seem to get down to the level of relations in objects..). What […]

units of group ring $\mathbb{Q}(G)$ when $G$ is infinite and cyclic

How would I be able to describe all units of the group ring $\mathbb{Q}(G)$ where $G$ is specifically an infinite cyclic group?

find a special element in group algebra

Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual convolution, i.e., $(\sum_{g\in G}\lambda_gg)(\sum_{h\in G}\mu_hh)=\sum_{g, h\in G}\lambda_g\mu_hgh$. Can we find $l=p_1(y, z)x^{n_1}+\cdots p_k(y,z)x^{n_k}\in l^1(G)^{\times}$ such that $\sum_{i=1}^k2^{n_i}p_i(y,z)(1-z^{n_i}y)=0$? Here, $\forall~ 1\leq i\leq k, ~p_i(y,z)\in \mathbb{Z}G$ […]

Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ is a nilpotent ideal ($p$ is a prime, $G$ is a $p$-group)

Let $p$ be a prime and let $G$ be a finite group of order a power of $p$ (i.e., a $p$-group). Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ (to be read as $\left( \mathbb{Z}/p\mathbb{Z} \right) G$) is a nilpotent ideal. (Note that this ring may be noncommutative.) Let $I_G$ be the augmentation […]

Torsion free abelian groups $G,H$ such that $k \cong k$ (as rings) for any field $k$

Let $G,H$ be torsion free abelian groups such that $k[G] \cong k[H]$ for any field $k$. Then is it true that $G \cong H$ ? If this is not true, then what if I change the hypothesis to $R[G]\cong R[H]$ for any non-zero commutative unital ring; is the conclusion true then ?

If $G$ is an infinite group, then the group ring $R(G)$ is not semisimple.

Let $R$ be a ring and $G$ an infinite group. Prove that $R(G)$ (group ring) is not semisimple. My idea was to suppose it is semisimple, then $R(G)$ is left artinian and $J(R(G))=0$. I was trying to make a ascending chain of ideals that won’t stop, then it is not left noetherian, by Hopkins theorem […]

How to recover the integral group ring?

I would like to solve the following exercise: Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times A_2$. Now $H_1$ and $H_2$ are finite $p$-groups, where $p \nmid \vert A_i \vert, i=1,2$. Prove that $RG_1 \cong RG_2$ iff $\mathbb Z A_{1} […]

Ring isomorphism for $k(G \oplus \mathbb Z )$ with $G$ torsion-free and abelian

Let $k$ be a field and $G$ be a torsion-free abelian group. Then $k[G]$ is an integral domain. If we denote its field of fractions by $F = k(G)$, is it true that $k(G \oplus \mathbb Z )\cong F(X)$? If not true in general , is it true if $k=\mathbb Q$ ?