Commutative rings whose proper ideals are serial
type: Item
instance of: scholarly article
title: Commutative rings whose proper ideals are serial
zbMATH Open document ID: 1391.13047
author: S. Heidari
author: M. Behboodi
published in: Algebras and Representation Theory
publication date: 2018-02-15T00:00:00Z
review text: Let \(R\) be a commutative ring with identity and \(M\) be an \(R\)-module. Then \(M\) is called uniserial, if its submodules form a chain under inclusion. Also \(M\) is called serial, when it is a direct sum of uniserial modules. If \(R\) is a uniserial (resp., serial) \(R\)-module, then it is said that \(R\) is a uniserial (resp., serial) ring. In this paper, commutative rings in which every proper ideal is serial are characterized. The authors show that if \((R,\mathfrak{M})\) is a local ring, then every proper ideal of \(R\) is serial, if and only if \(\mathfrak{M}\) is a direct sum of a uniserial module and a semisimple module. They also present examples of local rings in which every proper ideal is serial, but the ring itself is not serial. If \(R\) is not a local ring, the authors prove that every proper ideal of \(R\) is serial, if and only if \(R\) is a serial ring. Moreover, they show that in this case there exists an integer \(n\) such that every ideal of \(R\) is a direct sum of at most \(n\) uniserial \(R\)-module.
Mathematics Subject Classification ID: 13F99
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13A18
zbMATH DE Number: 6839441
zbMATH Keywords: serial modules
zbMATH Keywords: serial rings
zbMATH Keywords: uniserial modules
reviewed by: Ashkan Nikseresht
MaRDI profile type: Publication
full work available at URL: s10468-017-9699-7
OpenAlex ID: W2615618224
cites work: Q5801392
cites work: Q5563439
cites work: Commutative Noetherian local rings whose ideals are direct sums of cyclic modules
cites work: Commutative rings whose proper ideals are direct sum of completely cyclic modules
cites work: Commutative local rings whose ideals are direct sums of cyclic modules
cites work: Commutative rings with restricted minimum condition
cites work: Rings for which every module is a direct sum of cyclic modules
cites work: Q4318499
cites work: The structure of serial rings
cites work: Krull-Schmidt fails for serial modules
cites work: Uniserial modules over valuation rings
cites work: On the decomposition of modules and generalized left uniserial rings
cites work: On the structure of principal ideal rings
cites work: Q3166698
cites work: Elementary Divisors and Modules
cites work: Q5603302
cites work: Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring
cites work: Q4524978
cites work: Serial modules and endomorphism rings
cites work: Q5562681
cites work: Rings whose modules have nice decompositions
cites work: Serial rings and finitely presented modules
cites work: Q4002547
cites work: Q5732721
DOI: 10.1007/S10468-017-9699-7
description: scientific article; zbMATH DE number 6839441