AbstractLet R be an associative ring with unity. A unital left R-module M is said to be distributive if for every submodules S, T and U of M, the equality S∩(T+U)=S∩T+S∩U holds true. In this paper, we give a necessary and sufficient condition for a direct sum of left R-modules to be distributive. This condition is given by the notion of splitting of submodules of the direct sum and the proof uses the notion of orthogonality, where both notions are discussed and revisited.Keywords: Dedekind domaindistributive moduleorthogonalityPrüfer domainsplitting relative to the direct sum decompositionuniserial module2020 Mathematics Subject Classification: Primary: 13C05, 13C13Secondary: 13F05, 13C99 AcknowledgmentsThe authors would like to express their thanks to the referee for careful checking of the details of the paper and for pointing out the typos. 

A necessary and sufficient condition for a direct sum of modules to be distributive

On torsion elements and their annihilators

Abstract As a proper generalization of comultiplication modules, the concept of k-comultipli-cation semimodules is introduced in this paper. Let S be a semiring. An left S-semimodule M is called a left k-comultiplication semimodule if for each subtractive subsemimodule N of M, there exists an ideal I of S such that N=(0:MrI). Some properties of k-comultiplication semimodules are obtained and some conditions are given for a direct sum of k-comultiplication semimodules to be a k-comultiplication semimodule.

K
 -comultiplication semimodules

Abstract A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterization of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa. 

Classification of multiplication modules over multiplication rings with finitely many minimal primes

Criteria for a direct sum of modules to be a multiplication module over noncommutative rings

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Criteria for a direct sum of modules to be a multiplication module over noncommutative rings