# 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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### Partially commutative elements in powers of augmentation ideal

Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...
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### A question about group ring of a group over matrix ring!

Let $S$ be a ring and $G$ a group. We denote by $SG$ the group ring of $G$ over $S$. Let $S=M_n(R)$ be the set of all $n \times n$ matrices over a ring $R$. Is it true $SG\cong M_n(RG)$ (as rings)? ...
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### Find an example of a group algebra with non-trivial solutions to $x^2=x$.

I am looking for a group-algebra with a non-trivial solution to $x^2=x$. That is to say, a solution with $x\neq 1$ and $x \neq 0$ where $1$ is the identity. We have $x\subset \mathbb{C}[G]$ for some ...
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### Hom$_{i[k]}(k[G], R) \approx$ Hom$(G,R^*)$

Let $k$ be a field, $G$ a group and $R$ a $k$-algebra (i.e. a ring $R$ with a homomorphism $i : k \rightarrow Z(R)$). The claim is that there is a natural bijection between the set of $k$-algebra ...
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### GAP: how to work with elements of the group ring of the symmetric group $\mathbb{C}[S_k]$?

In GAP, working with elements of the symmetric group $S_k$ is straightforward. E.g. one can write (1,2)*(2,3); to obtain (1,3,2). Is there a similar ...
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### Decomposition of $C[G]$ as $C[G]$-module into direct sum of submodules of the form $Ce_{\chi}$

Let C be the complex field, and $G$ be a cyclic group generated by $a$. The group ring $C[G]$ is a $C[G]$-module (over itself) with the module action $C[G]\times C[G] \rightarrow C[G]$ the same as the ...
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### Group algebra is domain iff doesn't contain nonzero element whose square is 0

Let $G$ be a torsion-free group and let $K$ be a field. I have to prove that the group algebra $KG$ is an integral domain if and only if it doesn't contain a nonzero element whose square is equal to 0....
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### Group ring definition

I don't understand the definition of a group ring. In "An Introduction to Group Rings" by Polcino and Sehgal (page 129), an element of the group ring $RG$ is a linear combination of elements from $G$ ...
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