Finitely generated powers of prime ideals

TL;DR

This paper investigates conditions under which prime and maximal ideals with finitely generated powers are themselves finitely generated in various classes of commutative rings, extending Roitman's 2001 theorem.

Contribution

It extends Roitman's theorem by establishing new conditions for finitely generated prime ideals in rings like locally coherent, arithmetical, and polynomial rings.

Findings

Maximal ideals with finitely generated powers are finitely generated in locally coherent, arithmetical, polynomial rings.

Prime ideals with finitely generated powers are finitely generated in reduced coherent rings and polynomial rings over reduced arithmetical rings.

The results generalize and extend previous theorems on prime ideals in coherent integral domains.

Abstract

Let R be a commutative ring. If P is a maximal ideal of R whose a power is finitely generated then we prove that P is finitely generated if R is either locally coherent or arithmetical or a polynomial ring over a ring of global dimension $\le$ 2. And if P is a prime ideal of R whose a power is finitely generated then we show that P is finitely generated if R is either a reduced coherent ring or a polynomial ring over a reduced arithmetical ring. These results extend a theorem of Roitman, published in 2001, on prime ideals of coherent integral domains.