A maximal ideal wrt not containing a finitely generated ideal (D&F 7.4.35)

Let \(A = (a_1, \ldots, a_n)\) be a nonzero finitely generated ideal of \(R\). Prove that there is an ideal \(B\) which is maximal with respect to the property that it does not contain \(A\).

Let \(S\) be the set of ideals not containing \(A\). If \(C\) is a chain in \(S\), then define \(J = \cup_{I \in C} I\). Suppose that \(A \subseteq J\); then in particular each \(a_i\) is in some \(I_i \in C\). Thus the ideal \(I' = \cup_{1 \leq i \leq n} I_i\) contains each \(a_i\) and hence \(A\) itself. But this is a contradiction since \(I' \in C \subseteq S\). Thus \(J\) does not contain \(A\) and is an upper bound for \(C\). By Zorn's Lemma, there exists a maximal ideal \(B\) with respect to not containing \(A\).