On Nonnil-Noetherian Rings

Abstract

Let R be a commutative ring with 1 such that Nil(R) is a divided prime ideal of R. The purpose of this paper is to introduce a new class of rings that is closely related to the class of Noetherian rings. A ring R is called a Nonnil-Noetherian ring if every nonnil ideal of R is finitely generated. We show that many of the properties of Noetherian rings are also true for Nonnil-Noetherian rings; we use the idealization construction to give examples of Nonnil-Noetherian rings that are not Noetherian rings; we show that for each n ≥ 1, there is a Nonnil-Noetherian ring with Krull dimension n which is not a Noetherian ring.