On modules with DICC
type: Item
instance of: scholarly article
title: On modules with DICC
zbMATH Open document ID: 0611.13014
DOI: 10.1016/0021-8693(87)90074-3
author: Maria Contessa
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/26734
review text: A module M over a commutative ring R with unit is said to be DICC when it contains no doubly infinite chain of submodules and SDICC when it is DICC but neither ACC nor DCC. It is well-known that the existence of a faithful ACC module implies R is ACC, and a result of \textit{W. Heinzer} and \textit{D. Lantz} [J. Algebra 95, 201-216 (1985; Zbl 0588.13015)] states that, for R complete and quasi-local, the existence of a faithful DCC module implies R is ACC. The author gives an example to show that the existence of a faithful SDICC module does not imply R is DICC even when R is quasi-local and R/\({\mathfrak n}\) is complete (\({\mathfrak n}=rad 0)\). However the existence of a faithful SDICC module does imply R is weakly DICC, i.e. R/\({\mathfrak n}\) is Noetherian, \({\mathfrak n}\) is DCC and nilpotent. The author also adds to her characterizations of DICC rings and modules [J. Algebra 101, 489-496 (1986; Zbl 0589.13009) and 105, 429-436 (1987; Zbl 0606.13013)] a new characterization of SDICC.
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 3987422
zbMATH Keywords: doubly infinite chain condition
zbMATH Keywords: DICC
zbMATH Keywords: faithful SDICC module
reviewed by: David Kirby
MaRDI profile type: Publication
OpenAlex ID: W2090136772
cites work: On DICC rings
cites work: On rings and modules with DICC
cites work: Artinian modules and modules of which all proper submodules are finitely generated
cites work: Injective modules over Noetherian rings
cites work: Q5735274
description: scientific article; zbMATH DE number 3987422