Exercise 5, § 4

Let $E = A[X_1, \ldots, X_p]$, $F = A[[X_1, \ldots, X_p]]$. a) If D is a Z-derivation of E into F, we have $\omega(Du) \geq \omega(u) - 1$ for every non-zero $u$ in E. b) If $D'$ is a Z-derivation of F, then $\omega(D'v) \geq \omega(v') - 1$ for every non-zero $v$ in F. c) If E is equipped with the topology induced by that of F, then D and $D'$ are continuous. d) Every Z-derivation of E into F extends in a unique fashion to a Z-derivation of F. e) Let $\mathscr{D}$ be the F-module of A-derivations of F. If $D \in \mathscr{D}$ and $DX_i = u_i$, then

f) $(D_1, \ldots, D_p)$ is a basis of the F-module $\mathscr{D}$.