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Homological Objects of the Proper Class Determined by Weakly D-Closed Modules
* 1 , 2 , 3
1  Faculty of Sciences and Arts, Department of Mathematics, İpekköy Campuse, Amasya University, Amasya, 05100, Türkiye (Turkey)
2  Department of Engineering Sciences, Faculty of Engineering, Abdullah Gül University, Kayseri, 38100, Türkiye
3  Faculty of Sciences, Department of Mathematics, Atatürk University, Erzurum, 25240, Türkiye (Turkey)
Academic Editor: Irina Cristea

Abstract:

This talk is devoted to the study of a distinguished class of modules characterized by the property that all crumbling submodules of their factor modules are trivial. This defining condition imposes strong internal constraints on the module structure and leads naturally to a coherent and well-organized framework within module theory and relative homological algebra. The motivation behind this investigation arises from the observation that such a restriction allows for a systematic treatment of both structural and homological properties, thereby enabling meaningful comparisons with classical module theory concepts. The primary objective of the talk is to analyze the fundamental structural properties of this class of modules and to examine the relative classes that are naturally induced by it. Within this framework, the proposed module class is positioned among several well-known and extensively studied classes of modules in the literature. The relationships between these classes are explored in a precise and systematic manner, with particular attention given to similarities, differences, and inclusion properties. In particular, the effects of radicals, direct products, and module homomorphisms on the defining properties of the class are examined in detail. This analysis provides insight into the stability of the class under common operations and clarifies the extent to which the defining condition is preserved. Moreover, it sheds light on how these modules interact with classical methods and techniques in module theory. Overall, the results presented in this talk contribute to a deeper understanding of the interplay between structural properties and homological considerations in module theory. By developing and refining the conceptual framework surrounding this module class, the study lays the groundwork for further investigations in relative homological algebra and highlights new directions for research based on the interaction between algebraic constructions and module theory.

Keywords: a weakly D-closed module, proper class

 
 
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