Group Ring

In this page $R$ is a ring with unity $1_R$ and $G$ is a finite group.

Definition

$RG$ is a group ring if it holds the following conditions.

• $RG$ is a free $R$-module with basis $G$, i.e. $r_1{g}_1+\cdots +r_n{g}_n\in RG$.
• $r0=0\in RG$, $\forall r\in R$.
• $1=1e\in RG$.
• Multiplication $\left({\sum }_i{r}_i{g}_i\right)\cdot ({\sum }_j{r}_j^{\prime }{g}_j^{\prime })={\sum }_{i,j}{r}_i{r}_j^{\prime }{g}_i{g}_j^{\prime }$.

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Property

1. If $R$ is a nonzero ring and $G$ is a nontrivial finite group, $RG$ has zero divisors.

Proof

Let $p$ be a prime divisor of $\left|G\right|$. By the Cauchy’s theorem, there is a cyclic subgroup $C_p\le G$.
Let $a$ be a generator of $C_p$. Then $(1-a)(1+a+\cdots +a^{p-1})=0$. Therefore, $RG$ has a zero divisor $(1-a)$ and $(1+a+\cdots +a^{p-1})$.