Exercise 11, § 2

Let $p$ be a prime number, G a group that does not contain any elements of order $p, A$ a commutative ring of characteristic $p$ that does not contain any nonzero nilpotent elements, A[G] the algebra of G over A, and $(e_g)$ the canonical basis of A[G] over A. Prove that the ring A[G] has no nonzero nil ideal (observe that if an element $\sum a_ge_g\in A[G]$ is nilpotent, then we have $a_1= 0$).