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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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.
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States of a Group Ring

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on ...
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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 ...
Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
Is $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n\mathbb{Z}[G]$?
Let $G$ be a group, $\mathbb{Z}[G]$ be it's group ring and $G^n$ the direct product of $n$ copies of $G$. Is the group ring $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n \mathbb{Z}[G]$? If not, is it a ...