Corollary

— Assume that A is an infinite integral domain; let M be a free A-module and N a torsion-free A-module.

(i) We have $\mathrm{Pol}(M, N) = \bigoplus_{q \geq 0} \mathrm{Pol}^q(M, N)$ and each $\mathrm{Pol}^q(M, N)$ may be identified canonically with $\mathrm{Hom}(\mathbf{T S}^q(M), N)$.

(ii) Let $f \in \mathrm{Pol}(M, N)$ and $(e_i)_i$, a basis of M. There exists one and only one family $(u_\nu)_{\nu \in N^{(I)}}$ of elements of N such that $f \left( \sum_{i \in I} \lambda_i e_i \right) = \sum_{\nu \in N^{(I)}} \lambda^\nu u_\nu$ for all $(\lambda_i) \in A^{(I)}$.

The assertion (i) follows from Prop. 16 and 19, and (ii) follows from (i) and the Cor. of Prop. 16.