On commutative Noetherian rings which satisfy the radical formula

NONNIL-NOETHERIAN MODULES OVER COMMUTATIVE RINGS

In this paper we introduce a new class of modules which is closely related to the class of Noetherian modules. Let $R$ be a commutative ring with identity and let $M$ be an $R$-module such that $Nil(M)$ is a divided prime submodule of $M$. $M$ is called a Nonnil-Noetherian $R$-module if every nonnil submodule of $M$ is finitely generated. We prove that many of the properties of Noetherian modul...

Commutative Subdirectly Irreducible Radical Rings

A ring R is radical if there is a ring S (with unit) such that R = J (S) (the Jacobson radical). We study the commutative subdirectly irreducible radical rings and show that such a ring is noetherian if and only if is finite. We present a reflection of the commutative radical rings into the category of the commutative rings and derive a lot of examples of the subdirectly irreducible radical rin...

Superdecomposable pure injective modules over commutative Noetherian rings

We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are “tame” according to the Klingler–Levy analysis in [4], [5] and [6], in particular over Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases.

D-Bases for Polynomial Ideals over Commutative Noetherian Rings

On Nonnil-Noetherian Rings

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; ...