Exercise 24, § 1

Let G be a group, $e$ its identity element, and A a nonzero commutative ring. Denote the algebra of the group G over A by A[G] (III, §2, No. 6, p. 446). a) Let I be a finite set, $(G_i)_{i\in I}$ a family of subgroups of G, and $(g_i)_{i\in I}$ a family of elements of G. Prove that if G is the union of the family $(g_iG_i)_{i\in I}$, one of the subgroups $G_i$ has finite index in G.

b) Suppose that every conjugacy class in G distinct from $\{e\}$ is infinite and that the ring A is an integral domain. Prove that if $\mathfrak{a}$ and $\mathfrak{b}$ are two nonzero two-sided ideals of the ring A[G], we have $\mathfrak{a}\mathfrak{b}\not= 0$. (Choose elements $a=\sum a_gg$ and $b=\sum b_gg$ in $\mathfrak{a}$ and $\mathfrak{b}$ with $a_e\not= 0$ and $b_e\not= 0$. Using a), prove that, if necessary replacing $a$ with $hah^{-1}$ for suitable $h\in G$, we may assume that $a_gb_{g^{-1}}= 0$ for every $g\not=e$ in G.) $*$c) Prove that the ring A[G] is left (resp. right) Artinian if and only if the ring A is

Artinian and the group G is finite. (If A[G] is left Artinian, deduce from Theorem 1 of VIII, p. 6 that the group G has finite length; to prove that it is finite, reduce to the case when A is a field and G a simple group, and then use b) and Proposition 1 of VIII, p. 173$).*$