Exercise 26, § 3

Let A be a factorial ring, K its field of fractions. In order that elements f_i (1 ≤ i ≤ r) of B = A[X_1, ..., X_n] be such that the ideal $\sum_{i=1}^r Bf_i$ be equal to B, it is necessary and sufficient that there exist polynomials v_i (1 ≤ i ≤ r) of K[X_1, ..., X_n] such that $\sum_{i=1}^r v_i f_i = 1$ and which satisfy in addition the following condition: if one puts v_i = w_i/d, where d ∈ A, where the polynomials w_i belong to A[X_1, ..., X_n] and the g.c.d. of the set of the coefficients of all the w_i (1 ≤ i ≤ r) is equal to 1, then, for every extremal element p of A dividing d, the ideal generated by the classes of the f_i in the ring (A/Ap)[X_1, ..., X_n] is this ring in its entirety.