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 |
Externally published | Yes |
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