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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 557-573 (2001)
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Finite and Infinite Collections of Multiplication Modules

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Majid M. Ali, David J. Smith

Department of Mathematics, University of Auckland, Auckland, New Zealand, e-mail: majid@math.auckland.ac.nz; smith@math.auckland.ac.nz

**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.

**Keywords:** multiplication module, multiplication ideal, Pr\"{u}fer domain, arithmetical ring, radical, direct sum, prime submodule, residual, torsion module

**Classification (MSC2000):** 13C05; 13A15

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