Proposition 18

— *Let M, N, P be A-modules, and assume that M and N are free. If $f \in \mathrm{Pol}(M, N)$, $g \in \mathrm{Pol}(N, P)$, then $g \circ f \in \mathrm{Pol}(M, P)$.

We can reduce at once to the case where there exists an integer q such that $g \in \mathrm{Pol}^q(N, P)$; then there exists a q-linear mapping $h$ of $N^q$ into P such that $g(y) = h(y, y, \ldots, y)$ for all $y \in N$. Writing $f$ as a sum of homogeneous polynomial mappings we are thus reduced to proving that the mapping

of M into P, where $f_i \in \mathrm{Pol}^{q_i}(M, N)$, is polynomial. For $i = 1, \ldots, q$ there exists a $q_i$-linear mapping $l_i$ of $M^{q_i}$ into N such that $f_i(x) = l_i(x, x, \ldots, x)$ for all $x \in M$. Hence

from which our assertion follows.