Proposition 19

— Let M be a free A-module, N an A-module, G an infinite additive subgroup of A, and suppose that the homotheties of N defined by the nonzero elements of G are injective. Then Pol(M, N) is the direct sum of the Pol$^q$(M, N).

Let $f_0, f_1, \ldots, f_n$ be such that $f_i \in \mathrm{Pol}^i(M, N)$ and suppose that we have the relation $f_0 + \ldots + f_n = 0$. Let $x \in M$, then for all $\lambda \in G$ we have

By Lemma 2, applied to the polynomial $\sum_{i=0}^n f_i(x) X^i$ we have