Finite generation of powers of ideals

Abstract:

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) $R = K[X;\tilde S]$ is a monoid domain over a field $K$, where $\tilde S = S \cup \{0\}$ 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 $3 \le d \le \infty$, we establish existence of a $d$-dimensional integral domain having a nonfinitely generated maximal ideal $M$ of height $d$ such that $M^2$ is $3$-generated.