### 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 |
---|---|

Pages (from-to) | 557-573 |

Number of pages | 17 |

Journal | Beitrage zur Algebra und Geometrie |

Volume | 42 |

Issue number | 2 |

Publication status | Published - 2001 |

### Fingerprint

### 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

*Beitrage zur Algebra und Geometrie*,

*42*(2), 557-573.

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

Research output: Contribution to journal › Article

*Beitrage zur Algebra und Geometrie*, vol. 42, no. 2, pp. 557-573.

}

TY - JOUR

T1 - Finite and infinite collections of multiplication modules

AU - Ali, Majid M.

AU - Smith, David J.

PY - 2001

Y1 - 2001

N2 - 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.

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.

KW - Arithmetical ring

KW - Direct sum

KW - Multiplication ideal

KW - Multiplication module

KW - Prime submodule

KW - Prüfer domain

KW - Radical

KW - Residual

KW - Torsion module

UR - http://www.scopus.com/inward/record.url?scp=2942630349&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2942630349&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:2942630349

VL - 42

SP - 557

EP - 573

JO - Beitrage zur Algebra und Geometrie

JF - Beitrage zur Algebra und Geometrie

SN - 0138-4821

IS - 2

ER -