A Note On Noeherian 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 .