On A Certain Krull Symmetry Of a Noetherian Ring

TL;DR

This paper investigates the symmetry properties of Noetherian rings that are Krull-homogenous on both sides, establishing conditions under which certain prime ideals satisfy nilpotency and equality of specific prime sets.

Contribution

It demonstrates that for such rings, the intersection of certain prime ideals is nilpotent and the sets of primes defined by their Krull dimensions coincide, revealing a symmetry property.

Findings

P^n=0 for some n≥1

The sets Λ and v are equal

Prime ideals exhibit nilpotency and symmetry

Abstract

In this note we show that if a noetherian ring R is left and right Krull-homogenous and if: \Lambda ={P\textexclamdown {\epsilon} spec.R/ |R/P\textexclamdown|_r =|R|_r} and v ={Qj {\epsilon} spec.R| |R/Qj|l=|R|l} and P =\cap P\textexclamdown{\epsilon} \Lambda P\textexclamdown and Q = \cap Qj{\epsilon}VQj then the following hold true; (1) Pn =0, for some integer n\geq1 (2) \Lambda =v