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[SOLVED] Group Rings

Cbarker1

Active member
Jan 8, 2013
236
Dear Everyone,

Does every theorem that holds for finite group holds for ring groups? Why or Why not?


Thanks
Cbarker1
 

Olinguito

Well-known member
Apr 22, 2018
251
What do you mean by “ring groups”?
 

Cbarker1

Active member
Jan 8, 2013
236
A group ring defined as the following from Dummit and Foote:

Fix a commutative ring $R$ with identity $1\ne0$ and let $G=\{g_{1},g_{2},g_{3},...,g_{n}\}$ be any finite group with group operation written multiplicatively. A group ring, $RG$, of $G$ with coefficients in $R$ to be the set of all formal sum

$a_1g_1+a_2g_2+\cdots+a_ng_n$, $a_i\in R$, $1\le i\le n$
 

Olinguito

Well-known member
Apr 22, 2018
251
Well, note that $G$ is a group whereas $RG$ is a ring, so not every theorem about $G$ may be applicable to $RG$. For example, $G$ may be a cyclic group, but there is no such thing as a cyclic ring, so a theorem about cyclic groups may not make sense when applied to rings.

What you can say is that $RG$ contains a subring isomorphic to $R$, namely
$$\{a\cdot e_G:a\in R\}$$
as well as a subset which, with respect to multiplication, forms a group isomorphic to $G$, namely
$$\{1_R\cdot g:g\in G\}.$$