Exercise 1, § 9

Let K be a commutative field of characteristic $p, M$ a $p$-primary $\mathbf{Z}$-module, A the algebra of the group M over K, and $\mathfrak{m}$ the set of elements $\sum_ma_me_m$ of A (cf. III, §2, No. 6, p. 446) with $\sum_ma_m= 0$.

a) Prove that A is a local ring, with maximal ideal $\mathfrak{m}$, and that $\mathfrak{m}$ is a nil ideal (cf. VIII, p. 41, Exercise 10).

b) Suppose, moreover, that the homothety $p_M$ is surjective. Prove that we have $\mathfrak{m}=\mathfrak{m}^2$.