Finite and infinite collections of multiplication modules

Majid M. Ali, David J. Smith

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

All rings are commutative with identity and all modules are unitary. In this note we give some properties of a finite collection of submodules such that the sum of any two distinct members is multiplication, generalizing those which characterize arithmetical rings. Using these properties we are able to give a concise proof of Patrick Smith's theorem stating conditions ensuring that the sum and intersection of a finite collection of multiplication submodules is a multiplication module. We give necessary and sufficient conditions for the intersection of a collection (not necessarily finite) of multiplication modules to be a multiplication module, generalizing Smith's result. We also give sufficient conditions on the sum and intersection of a collection (not necessarily finite) for them to be multiplication. We apply D. D. Anderson's new characterization of multiplication modules to investigate the residual of multiplication modules.

Original languageEnglish
Pages (from-to)557-573
Number of pages17
JournalBeitrage zur Algebra und Geometrie
Volume42
Issue number2
Publication statusPublished - 2001

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Multiplication Module
Multiplication
Intersection
Ring
Sufficient Conditions
Distinct
Necessary Conditions
Module
Theorem

Keywords

  • Arithmetical ring
  • Direct sum
  • Multiplication ideal
  • Multiplication module
  • Prime submodule
  • Prüfer domain
  • Radical
  • Residual
  • Torsion module

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

Cite this

Finite and infinite collections of multiplication modules. / Ali, Majid M.; Smith, David J.

In: Beitrage zur Algebra und Geometrie, Vol. 42, No. 2, 2001, p. 557-573.

Research output: Contribution to journalArticle

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