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- Thread starter Cbarker1
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- Apr 22, 2018

- 251

What do you mean by “ring groups”?

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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

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

- Apr 22, 2018

- 251

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

$$\{1_R\cdot g:g\in G\}.$$