Summary: Suppose \(M\) is a maximal ideal of a commutative integral domain \(R\) and that some power \(M^n\) of \(M\) is finitely generated. We show that \(M\) is finitely generated in each of the following cases:
(i) \(M\) is of height one,
(ii) \(R\) is integrally closed and \(\operatorname{ht} M=2\),
(iii) \(R = K[X;\widetilde S]\) is a monoid domain over a field \(K\), where \(\widetilde S =S \cup \{0\}\) is a cancellative torsion-free monoid such that \(\bigcap_{m=1}^\infty mS=\emptyset\), and \(M\) is the maximal ideal \((X^s:s\in S)\).
We extend the above results to ideals \(I\) of a reduced ring \(R\) such that \(R/I\) is Noetherian. We prove that a reduced ring \(R\) is Noetherian if each prime ideal of \(R\) has a power that is finitely generated. For each \(d\) with \(3 \leq d \leq \infty\), we establish the existence of a \(d\)-dimensional integral domain having a nonfinitely generated maximal ideal \(M\) of height \(d\) such that \(M^2\) is \(3\)-generated.