Finite and infinite collections of multiplication modules

Majid M. Ali*, David J. Smith

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 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

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

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