Almost uniserial rings and modules.
type: Item
instance of: scholarly article
title: Almost uniserial rings and modules.
zbMATH Open document ID: 1341.16001
author: S. Roointan-Isfahani
author: M. Behboodi
published in: Journal of Algebra
publication date: 2015-11-17T00:00:00Z
review text: Generalizing the notions of uniserial modules and uniserial rings, the authors in this paper, introduce the notions of almost uniserial modules and almost uniserial rings and investigate their properties. They define an \(R\)-module \(M\) to be almost uniserial if any two non-isomorphic submodules of \(M\) are comparable and a ring \(R\) is said to be almost left (resp. right) uniserial if \(R\), as a left (resp. right) \(R\)-module, is almost uniserial. They prove that an almost left uniserial ring \(R\) is either local or every maximal left ideal of \(R\) is cyclic. From this they deduce that every Noetherian, almost left uniserial ring is either a local ring or a principal left ideal ring. They also prove that a left Artinian, principal left ideal ring \(R\) is almost left uniserial if and only if either \(R\) is left uniserial or \(R\) is isomorphic to the matrix ring \(R=M_2(D)\) for some division ring \(D\). The authors conclude the paper with a structure theorem for commutative Artinian almost uniserial rings.
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16P70
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 6509815
zbMATH Keywords: almost uniserial modules
zbMATH Keywords: almost uniserial rings
zbMATH Keywords: principal left ideal rings
zbMATH Keywords: commutative Artinian rings
reviewed by: Veereshwar A. Hiremath
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.09.024
OpenAlex ID: W2211380628
cites work: Q5801392
cites work: On left Köthe rings and a generalization of a Köthe-Cohen-Kaplansky theorem
cites work: Rings for which every module is a direct sum of cyclic modules
cites work: The structure of serial rings
cites work: Krull-Schmidt fails for serial modules
cites work: Uniserial modules over valuation rings
cites work: Non-commutative principal ideal rings
cites work: On the decomposition of modules and generalized left uniserial rings
cites work: Elementary Divisors and Modules
cites work: Local rings with bounded ideals
cites work: Rings with the dual of the isomorphism theorem.
cites work: Weak Krull-Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension.
cites work: A Version of the Weak Krull–Schmidt Theorem for Infinite Direct Sums of Uniserial Modules
cites work: Artinian and Noetherian serial rings
cites work: Some model theory over a nearly simple uniserial domain and decompositions of serial modules
cites work: SOME MODEL THEORY OVER AN EXCEPTIONAL UNISERIAL RING AND DECOMPOSITIONS OF SERIAL MODULES
cites work: Noncommutative generalizations of theorems of Cohen and Kaplansky.
cites work: Serial modules and endomorphism rings
cites work: Rings whose modules have nice decompositions
cites work: Serial rings and finitely presented modules
DOI: 10.1016/J.JALGEBRA.2015.09.024
description: scientific article; zbMATH DE number 6509815