Krull's intersection theorem in case of small submodules
type: Item
instance of: scholarly article
title: Krull's intersection theorem in case of small submodules
zbMATH Open document ID: 0808.13002
DOI: 10.1007/BF01201779
published in: Archiv der Mathematik
publication date: 1995-03-09T00:00:00Z
review text: Let \(R\) denote a commutative Noetherian ring and \(M\) an \(R\)-module. A submodule \(U\) of \(M\) is called small in case \(U + X \subsetneqq M\) for each submodule \(X \subsetneqq M\). It is common that the sum of all small submodules of \(M\) is the Jacobson radical \(\text{rad} M\) of \(M\) and that finitely generated submodules of \(\text{rad} M\) are small. In general, small submodules need not be finitely generated. However, it is shown in the present article that under reasonable assumptions small submodules share an important finiteness property with the finitely generated ones, namely the Krull intersection theorem. For instance, the latter holds for small submodules of torsion-free modules in case \(R\) is an integral domain. A consequence of this result is that over a local ring flat small submodules are finitely generated. Two applications of the finiteness conditions for small submodules are given. First, they are used for a very explicit description of the set of coassociated prime ideals of a flat module \(M\). (These are the prime ideals of \(R\) which are annihilators of artinian factor modules of \(M\).) -- Second, conditions are given under which rings between \(R\) and its total quotient ring \(K\) are small submodules of the \(R\)-module \(K\). It is shown that the integral closure \(R'\) of \(R\) in \(K\) has this property, iff each maximal ideal of \(R\) is regular, and conversely that a ring \(R \subset A \subset K\) must be contained in \(R'\) provided that it is small in \(K\) and \(A/{\mathfrak m} A\) is a finitely generated \(R\)-module for each maximal ideal \({\mathfrak m}\) of \(R\).
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13A10
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13C11
zbMATH DE Number: 563363
zbMATH Keywords: small submodules
zbMATH Keywords: Jacobson radical
zbMATH Keywords: Krull intersection theorem
zbMATH Keywords: coassociated prime ideals of a flat module
zbMATH Keywords: integral closure
author: Helmut Zoeschinger
MaRDI profile type: Publication
cites work: Q4117922
cites work: Q3739243
full work available at URL: https://doi.org/10.1007/bf01201779
OpenAlex ID: W1681693167
Recommended article: Modules whose small submodules have Krull dimension
Recommended article: Q5014360
Recommended article: Q4205470
Recommended article: On the small Krull dimension of modules
Recommended article: Summand intersection property on the class of exact submodules
Recommended article: An Elementary Proof of Krull's Intersection Theorem
Recommended article: Small submodules relative to a torsion theory
Recommended article: A short proof of Krull's intersection theorem
Recommended article: ON SMALL-SEMIESSENTIAL SUBSEMIMODULES
Recommended article: On generalizations of essential and small submodules
description: scientific article; zbMATH DE number 563363