On rings and modules with DICC
type: Item
instance of: scholarly article
title: On rings and modules with DICC
zbMATH Open document ID: 0589.13009
DOI: 10.1016/0021-8693(86)90207-3
author: Maria Contessa
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/26128
review text: A module over a commutative ring R has the property DICC if every chain of submodules indexed by the integers terminates to one side or the other. The author characterizes such modules as those modules M having an Artinian submodule N such that M/N is Noetherian and N/Rx\(\cap N\) is of finite length for all \(x\in M\setminus N\). A construction of rings that have DICC as modules over themselves is given. It is shown that if M has DICC, but is neither Noetherian nor Artinian, and if no maximal ideal is minimal in the support of M, then the support of M is closed and irreducible in Spec(R) Some of the proofs use an interesting concept called almost-divisibility. The R-module Q is almost-divisible if Q/aQ has finite length, for all a not in the nilradical of R.
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13A05
zbMATH DE Number: 3946298
zbMATH Keywords: DICC
zbMATH Keywords: chain of submodules
zbMATH Keywords: Artinian submodule
zbMATH Keywords: almost-divisibility
reviewed by: William H. Gustafson
MaRDI profile type: Publication
OpenAlex ID: W1971644269
cites work: Artinian modules are countably generated
cites work: Q5519970
cites work: Injective modules over Noetherian rings
cites work: Modules With Descending Chain Condition
cites work: The Dual of the Notion of “Finitely Generated”
description: scientific article; zbMATH DE number 3946298