Exercise 2, § 21

Let A be a commutative ring and G a group.

a) Suppose that the algebra A[G] of the group G is an absolutely flat ring (VIII, p. 171, Exercise 27). Prove that the ring A is absolutely flat. Let H be a subgroup of G that admits a finite generating subset; prove that the left ideal $I_H$ of A[G] (VIII, p. 22, Exercise 25) admits a supplementary in A[G], and deduce that H is finite (observe that in the opposite case, the annihilator of $I_H$ would be reduced to (0)).

b) Keep the assumption of a). Let $g$ be an element of G of finite order $n$. Prove that $n1_A$ is invertible in A (let $x$ be an element of A such that $(1-e_g)x(1-e_g) = (1-e_g)$; construct an element $y$ of A such that $1-(1-e_g)x=y(1 +e_g+\cdots +e_{g^{n-1}})$). Deduce that the order of every finite subgroup of G is invertible in A.

c) Suppose that A is an absolutely flat ring and that every finite subset of G generates a finite subgroup with cardinal invertible in A. Prove that the ring A[G] is absolutely flat (reduce to the case when G is finite, and use Exercise 1).