# 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.

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### Group ring confusion

This actually causes a lot of confusion. For a finite group $G$ and a commutative unit ring $R$ I’m trying to prove that $h\in R[G]$ defined as $h=\sum\limits_{g\in G}g$ is in the center of $R[G]$. ...
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### Group algebra functor preserves colimits

Consider a commutative unital ring $R$ and the group algebra functor $$R[-]:\bf{Grp}\rightarrow {Alg}_R$$ which has the group of units functor $$(-)^\ast:\bf{Alg}_R\rightarrow \bf{Grp}$$ as right-...
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### About the matrix representation of group algebra

Consider the group algebra $\mathbb{R}[C_3]$，where $\mathbb{R}$ is real field and $C_3$ is $3$-order cyclic group. It's known $C_3$ can be represented as $\{1,e^{2\pi i/3},e^{4\pi i/3}\}$, I tried to ...
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### What breaks if I use a $G$-module instead of a $\mathbb{K}[G]$-module: Induced reps, Frobenius reciprocity?

The Question I use $\mathbb{K}G$-module to denote a $G$-action on a vector space over $\mathbb{K}$ (side question - is this the standard notation?). A $\mathbb{K}[G]$-module differs in that we allow ...
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### 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 $k(G \oplus \mathbb Z )\cong F(X)$? ...
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### Simple modules of the group algebra $kC_p$ over different fields $k$

I want to find the simple modules for the group algebra $A=kC_7$, where $k$ is a field and $C_7$ is the cyclic group of order $7$. When $k = \mathbb{C},$ by Maschke's Theorem $A$ must be semisimple. ...
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### Intuition behind restriction and extension of scalars of group rings

Suppose we have a finite group $G$ with subgroup $H\leq G$. If $R$ is a commutative ring then we have the group rings $B=R[G]$ and $A=R[H]$, along with the natural inclusion $i:A\hookrightarrow B$. ...
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### Generalized Formal Power Series Ring

I came up with the following generalization of formal power series and wonder if anyone knows a reference where this is studied. Let $R$ be a commutative ring with unity and $M$ a commutative monoid ...
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### Simple notation question: What does $(kG)^\times$ mean? $k$ is a field and $G$ is a group

I guess $kG$ is the group algebra over a finite group, i.e. the set of linear combinations of $k$ and group elements. But I do not understand the "$^\times$". Thank you.