Finite generation of powers of ideals

Suppose M is a maximal ideal of a commutative integral domain R and that some power Mⁿ 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 ht M = 2, (iii) R = K[X;S̃] is a monoid domain over a field K, where S̃ = S ∪ {0} is a cancellative torsion-free monoid such that ∩_m=1^∞ mS = 0, and M is the maximal ideal (X^s : s ε 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 ≤ d ≤ ∞, we establish existence of a d-dimcnsional integral domain having a nonfinitely generated maximal ideal M of height d such that M² is 3-generated.