A Note On Noeherian Rings

TL;DR

This paper introduces the concepts of noetherian disjoint and non-disjoint rings, and proves properties of their nilradicals and semiprime ideals, expanding understanding of ring structures in algebra.

Contribution

It defines new classes of noetherian rings and establishes key properties of their nilradicals and semiprime ideals, providing a theoretical framework for these structures.

Findings

Nilradical is weakly ideal invariant in disjoint rings.

Equivalence of conditions for non-disjoint rings involving semiprime ideals.

Characterization of nilradical properties in noetherian rings.

Abstract

In this paper we introduce the definition of a noetherian disjoint ring and that of a noetherian non-disjoint ring . For a noetherian ring R , with nilradical N if P and Q represent the semiprime ideals of R called as the right and the left krull-homogenous parts of N as defined in [8] , then we prove the main theorem of this paper for the ring R whose statement is given below. Main Theorem :- Let R be a Noetherian ring with nilradical N . Let P and Q represent the right and the left krull-homogenous parts of N . Then the following hold true for the ring R ; (a) If R is a disjoint ring , then the nilradical N of R is a right and a left weakly ideal invariant ideal of R . Hence N is a right and a left localizable semiprime ideal of R . (b) If R is a non-disjoint ring then the following are equivalent conditions on R ; (i) N is a right and a left weakly ideal invariant ideal of R . (ii) P = Q is a right and a left localizable semiprime ideal of R .