# Question about a group algebra being not Artinian

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?