Example 4

Let $(X_i)_{i \in I}$ be a family of indeterminates, $N$ an $A$-module and $u \in N[(X_i)_{i \in I}]$ a homogeneous polynomial of degree $q$. The mapping $(x_i)_{i \in I} \mapsto u((x_i)_{i \in I})$ of $A^{(I)}$ into $N$ is a homogeneous polynomial mapping of degree $q$: this is seen at once by condition (iii) of Prop. 13. If $I$ is finite, every homogeneous polynomial mapping of degree $q$ of $A^{(I)} = A'$ into $N$ is of that form.