Mathematics Stack Exchange
Mathematics Stack Exchange

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.

The group ring $R[G]$ is constructed in the following way. The set $R[G]$ is the free $R$-module on the elements of $G$, equipped with the multiplication given by the operation in $G$ extended distributively to all elements in the free module.

A special case of this construction is group algebra, which occurs naturally in representation theory. It turns out that every group representation $\rho:G\rightarrow GL(V)$ corresponds to an $R[G]$ module structure on $V$. This connection ties the representation theory of groups to the module theory of group algebras.

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Prime ideals in group ring

Let $R$ be a ring of characteristic $p$ (can assume $R=\mathbb{F}_q$ if it makes things significantly easier) and $G$ be a finite group. Do we know anything about the prime ideals in $R[G]$? I've tried multiple searches including terms like "prime…
Daniel M
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Is the group ring of a finite cyclic group a "PID"?

Let $G=\langle\sigma\rangle$ be a group wirtten multiplicatively. Assume that $|G|=n$. The group ring of $G$ (denoted by $\mathbb{Z}[G]$) is defined as $$\mathbb{Z}[G]=\{\sum_{i=0}^{n-1}a_ig^i\text{ }|\text{ }a_i\in\mathbb{Z}\}$$ The sum of two…
learning_math
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Conceptual doubt in a theorem in Group rings

Let R be a ring and G be a group and $H \unlhd G$ , where RG is the group ring. Define $\widehat{H}= \sum_{h \in H } h$. And $e_H$ defined below is central idempotent. Now I was understanding the following proof.. Now I have some problem in…
Bhaskar Vashishth
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Description of the group $1+J(FG),$ where $J(FG)$ is jacobson radical of the group ring $GF.$

My group is $G=(\mathbb{Z}_3\times\mathbb{Z}_3)\rtimes\mathbb{Z}_3$ which is non abelian group of order $27.$ Now my problem is whether the group $1+J(FG)$ is abelian or non-abelian and what is its exponent? Here $F$ is any finite field of…
neelkanth
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Concerning subrings of a integral group ring

Let $G$ be a group . Does a subring of the integral group ring $\mathbb{Z}[G]$ has the form $\mathbb{Z}[H]$ for a subgroup $H$ of $G$? Thanks in advance.
M.Ramana
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how to prove intersection of normal subgroups controls the ideal

I have to prove the following: Let $I$ be an ideal of $K[G]$ ($K$ is a field, $G$ is a multiplicative group, $K[G]$ is a group ring) and let $H_1,\ldots,H_n$ be normal subgroups of $G$, each controlling $I$. Show that $H= H_1\cap H_2\cap\ldots\cap…
m15
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Unit group of $\mathbb{Z}C_m$

By Higman's theorem we know that if $G$ is abelian and finite then $\mathbb{Z}G$ has only trivial units iff G is has exponent $1,2,3,4, \text{or}\ 6$ But what if $G=C_{m}$ where $m \ge 7$. In this case has someone calculated what the…
Bhaskar Vashishth
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Two beginner questions about group rings

Let $RG$ denote a group ring and let $\epsilon: RG \to R $ be the augmentation map. Then $\ker(\epsilon)$, denoted $\Delta(G)$, is called augmentation ideal. If we let $\alpha= \sum_ {g \in G} a_g.g $ be an element of $\Delta(G)$ then it can be…
Bhaskar Vashishth
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$\mathbb{Z}_{2}G$ as the class of finite subsets of $G$

Let $G$ be a group. Then the group ring $RG$ is defined as $\sum_{g\in G} R$ (a copy of $R$ for every $g\in G$) and a typical element is of the form $ \sum_{i=1}^{n} r_{g_{i}}g_{i} $. Now if $R= \mathbb{Z}_{2}$ we can think $\mathbb{Z}_{2}G$ as the…
Teplotaxl
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${I}=\Delta(G,G')$ is the smallest ideal of the group ring $\mathbb{Z}{G} $ such that $\mathbb{Z}{G}/{I}$ is a commutative ring

How can I prove that if ${G}$ is a group then ${I}=\Delta(G,G')$ is the smallest ideal of the group ring $\mathbb{Z}{G} $ such that $\mathbb{Z}{G}/{I}$ is a commutative ring? It is an ideal is clear, as it is the kernel of homomorphism…
Bhaskar Vashishth
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Isomorphism problem over noncommutative rings.

This conjecture (over integeral group rings) states $$\Bbb{Z}G\cong \Bbb{Z}H \implies G\cong H$$. It can very well be studied over various commutative rings and people do that fixing some commutative ring and finding all classes of group that can be…
Bhaskar Vashishth
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