Corollary 1

Let $m$ be the ideal of polynomials $u \in A[X]$ such that $u(a) = 0$. Then $m$ is generated by the polynomials $X_i - a_i$ (for $i \in I$).

It is clear that $X_i - a_i \in m$ for each $i \in I$. Let $u \in m$ and let $v$ be as in Prop. 2. Since $v$ has no constant term, there exists a family $(P_i)_{i \in I}$, of polynomials in $A[X]$ with finite support such that

If we replace $X_i$ by $X_i - a_i$ for each $i \in I$ in the above equation, we obtain a relation of the form $u(X) = \sum_{i \in I} (X_i - a_i) \cdot P'_i(X)$, whence the corollary.