Remark

(1) Suppose that $E$ is a graded $A$-algebra of type $\mathbf{Z}$, with graduation $(E_n)$, and suppose also that the linear mapping $f$ (assumed to satisfy (1)) is such that (2)

Then the relation $g(x_1 x_2 \ldots x_p) = f(x_1) f(x_2) \ldots f(x_p)$ with the $x_i \in M$ shows that $g(S^p(M)) \subset E_p$ for all $p \geq 0$ and hence $g$ is a graded algebra homomorphism.

(2) Every element of $S(M)$ is a sum of products of the form $x_1 x_2 \ldots x_n$, where the $x_i$ belong to $M$; care should be taken not to confuse such products taken in $S(M)$ with the analogous products taken in $T(M)$.

(3) If $n!.1$ is invertible in $A$, the $A$-module $S^n(M)$ is generated by the elements of the form $x^n$, where $x \in M$; this follows from the above remark and I, § 8, no. 2, Proposition 2.