Exercise 26, § 3

Let $A$ be a factorial domain and $K$ its field of fractions. For elements $f_i$ ($1 \leq i \leq r$) of $B = A[X_1, \ldots, X_n]$ to be such that the ideal $\sum_{i=1}^r Bf_i$ is equal to B, it is necessary and sufficient that there exist polynomials $v_i$ ($1 \leq i \leq r$) in $K[X_1, \ldots, X_n]$ such that $\sum_{i=1}^r v_i f_i = 1$ and which satisfy further the following condition: if we write $v_i = w_i / d$, where $d \in \mathbf{A}$, the polynomials $w_i$ belong to $\mathbf{A}[X_1, \ldots, X_n]$ and the g.c.d. of the set of coefficients of all the $w_i$ ($1 \leq i \leq r$) is equal to 1, then, for every extremal element $\mathfrak{a}$ of $\mathbf{A}$ dividing $d$, the ideal generated by the classes of theft in the ring $(\mathbf{A}/\mathfrak{A}\mathfrak{p})[X_1, \ldots, X_n]$ is the whole of this ring.