Remark

— Let $x \in A^I$ and let $f$ be the mapping $u \mapsto u(x)$ of $A[X]$ into $A$. Given an $A$-module $M$, we consider the homomorphism $1 \otimes f$ of $M[X] = M \otimes_A A[X]$ into $M \otimes_A A = M$. For each $v \in M[X]$ we have $(1 \otimes f)(v) = v(x)$. If $v = \sum_{\nu \in \mathbf{N}^{(I)}} e_\nu X^\nu$, then $v(x) = \sum_{\nu \in \mathbf{N}^{(I)}} x^\nu e_\nu$.