Remark

— 2) The composite homomorphism $A, = u \circ 0 : S(M^*) \to \mathrm{Pol}_A(M, A)$ is the unique unital homomorphism of algebras inducing the inclusion of

in $\mathrm{Pol}(M, A)$. If $M$ is finitely generated free and $A$ is an infinite integral domain, then $\lambda_M$ is bijective (Prop. 20 and Cor. of Prop. 19). In particular if $A$ is an infinite integral domain, then the canonical homomorphism $f \mapsto \tilde{f}$ of $A[X_1, \ldots, X_n]$ into $\mathrm{Pol}(A^n, A)$ (IV, p. 4) is an isomorphism.

3) Consider the coproduct $c_S : S(M^*) \to S(M^* \times M^*)$ (III, p. 575, Example 6). For any $v \in S(M^*)$, $x, y \in M$, the polynomial mapping $\lambda_{M \times M}(c_S(v)) : M \times M \to A$ maps $(x, y)$ to $A, (v)(x + y)$. For the two algebra homomorphisms of $S(M^*)$ into $\mathrm{Map}(M \times M, A)$ defined in this way agree on $M^*$, in virtue of the relation