Finite generation of powers of ideals

Abstract

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