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Let $F=GF(2)$ and $G=\langle a_1,a_2,\ldots|a_i^3=1,[a_i,a_j]=1\text{ for }i,j\in\mathbb{N}\rangle$, that is the direct product of infinitely many cyclic groups of order $3$. I want to show that the group algebra $FG$ seen as an $FG$-module is not Artinian (so that I have $FG$ being not semisimple).

My attempt goes as follows. I take $1+a_1$ and I want to show that $(1+a_1)\supset((1+a_1)(1+a_2))$ as $FG$-modules. If this is true the process can be iterated and $FG$ is not Artinian. So, assume by a contradiction that there exists a $k$ in $FG$ such that $1+a_1=k(1+a_1)(1+a_2)$. Now I would say that this equation forces $k$ to belong to the $F(\langle a_1\rangle\times\langle a_2\rangle)$ and this cannot be.

Now my questions are: 1) Is my approach right/meaningful? 2) Are there best ways to show that $FG$ is not Artinian?

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