# Finite and infinite collections of multiplication modules

Majid M. Ali*, David J. Smith

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 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 language English 557-573 17 Beitrage zur Algebra und Geometrie 42 2 Published - 2001 Yes

## Keywords

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