On some conditions on a Noetherian ring

TL;DR

This paper investigates conditions under which the nilradical of a Noetherian ring exhibits weak ideal invariance, linking it to the localizability of certain semiprime ideals, and establishes their equivalence.

Contribution

It introduces the concepts of left and right Krull homogeneous parts of the nilradical and proves their equivalence to weak ideal invariance and localizability conditions.

Findings

Nilradical's weak ideal invariance is equivalent to localizability of semiprime ideals.

Defines left and right Krull homogeneous parts of the nilradical.

Establishes a main theorem linking invariance and localizability conditions.

Abstract

In this paper for a noetherian ring R with nilradical N we define semiprime ideals P and Q called as the left and right krull homogenous parts of N . We also recall the known definitions of localisability and the weak ideal invariance (w.i.i for short ) of an ideal of a noetherian ring R . We then state and prove results that culminate in our main theorem whose statement is given below ; Theorem :- Let R be a noetherian ring with nilradical N . Let P and Q be semiprime ideals of R that are the right and left krull homogenous parts of N respectively . Then the following conditions are equivalent ; (i) N is a right w.i.i ideal of R ( respectively N is a left w.i.i ideal of R ) . (ii)P is a right localizable ideal of R ( respectively Q is a left localizable ideal of R ) .