Representation type of commutative Noetherian rings II: Local tameness

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 Local Rings of bounded module type

Let R be a local ring of bounded module type. It is shown that R is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is an almost maximal valuation domain. We deduce from this that R is almost maximal if one of the following conditions is satisfied: R is a Q-algebra of Krull dimension ≤ 1 or the maximal ideal of R is the union of all non-maximal prime i...

Representation of real commutative rings

During the last 10 years there have been several results on the representation of real polynomials, positive on some semi-algebraic subset of R. These results started with a solution of the moment problem by Schmüdgen for corresponding sets. Later Wörmann realized that the same results could be obtained by the so-called “Kadison-Dubois” Representation Theorem. The aim of our talk is to present ...

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