### 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 |
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Pages (from-to) | 557-573 |

Number of pages | 17 |

Journal | Beitrage zur Algebra und Geometrie |

Volume | 42 |

Issue number | 2 |

Publication status | Published - 2001 |

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### 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.