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

### Fingerprint

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
• 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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AB - 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.

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