Corollary

The A-module $S^n(M)$ is the image under isomorphism (10) of the submodule of $\bigotimes_{\lambda \in L} S(M_\lambda)$ the direct sum of the submodules $S^\nu(M)$ for all the families $\nu = (n_\lambda) \in \mathbf{N}^{(L)}$ such that $\sum_{\lambda \in L} n_\lambda = n$; if $J$ is the support of $\nu$, $S^\nu(M)$ is canonically isomorphic to $\bigotimes_{\lambda \in J} S^{n_\lambda}(M_\lambda)$.

In general $S^\nu(M)$, $\bigotimes_{\lambda \in J} S^{n_\lambda}(M_\lambda)$ and their image in $S^n(M)$ are identified.