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 htM = 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 = ∅, and M is the maximal ideal (Xs : s ∈ 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-dimensional integral domain having a nonfinitely generated maximal ideal M of height d such that M2 is 3-generated.