Exercise 4, § 1

a) Suppose that for every non-zero $n \in \mathbf{Z}$, the relation $n\xi = 0$ implies $\xi = 0$ in A. Show that the mapping $f \mapsto f(D_1, ..., D_n)$ is an injective homomorphism of the polynomial algebra $A[X_1, ..., X_n] = E$ into the algebra (over A) of endomorphisms of the A-module E (use induction on n). b) Let m be an integer $\geqslant 1$ such that $m \cdot 1 = 0$ in A. Show that we have $D_i^m = 0$ for every partial derivation $D_i$ ($1 \leqslant i \leqslant n$) in $A[X_1, ..., X_n]$.