Locally noetherian commutative rings

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...

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

Fully Bounded Noetherian Rings

Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover resul...

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; ...