Prime Ideals in Noetherian Rings

TL;DR

This paper investigates the properties of prime ideals in Noetherian rings, focusing on links and extensions in polynomial rings, and introduces the concept of link Krull symmetry.

Contribution

It defines link Krull symmetric Noetherian rings and proves new theorems about the extension of linked prime ideals in polynomial rings.

Findings

Linked prime ideals in R[X] extend similarly under certain conditions

Theorem 9 establishes the extension property for linked primes in polynomial rings

Application to fully bounded Noetherian rings demonstrates broader relevance

Abstract

In this short note we study the links of certain prime ideals of a noetherian ring R. We first give the definition of a link krull symmetric noetherian ring R. We then prove theorem 9 that states that for any linked prime ideals P' and Q' of the polynomial ring R[X] where R is a link krull symmetric noetherian ring, if The prime ideal P' is extended then Q' is also an extended prime ideal of R[X]. An application of theorem 9 is then given in theorem 12 for the ring R[X] when R is assumed to be a fully bounded noetherian ring.