The Krull intersection theorem. II.

1. Introduction. This paper is a continuation of [1]. In §2 we show that for a nonminimal principal prime (p), /= Π* =1(p)" is a prime ideal and pJ = /. An example is given to show that the condition that (p) be nonminimal is necessary. We also consider the question of when a prime ideal minimal over a principal ideal has rank one. Of particular interest is the example of a domain D with a doubly generated ideal / such that Π;=1 Γϊl Π; βl Γ. In §3 we prove that Π: =1 ΓA = lΓ\™=ιΓA for any finitely generated module A over a valuation ring. In §4 we consider certain converses to the usual Krull Intersection Theorem for Noetherian rings. It is shown that for (R, M) a quasi-local ring whose maximal ideal M is finitely generated, many classical results for local rings are actually equivalent to the ring R being Noetherian.