Browsing by Author "Ssevviiri, David"

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    Characterizations of regular modules
    (International Electronic Journal of Algebra, 2022-12-14) Kimuli, Philly Ivan; Ssevviiri, David
    Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.
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    Modules with reduced endomorphism rings
    (World Scientific Publishing, 2024-10-17) Kimuli, Philly Ivan; Ssevviiri, David
    In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including K-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings R for which every faithful multiplication R-module is endo-reduced is precisely that of reduced rings; while the class of rings R for which every multiplication R-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.