SA-Extending and WS-Rationally Extending Modules: Structure and Duality
Keywords:
SA-extending modules; $\oplus$-s-extending modules; WS- rationally extending modules; SA-supplement modules; Lifting modules.Abstract
We present a comprehensive study of several generalized module classes, including SA-extending, $\oplus$-s-extending, SA-supplement, and WS-rationally extending modules. Detailed definitions, fundamental properties, illustrative examples, and structural results are provided to clarify the relationships among these classes. Dual notions, such as SA-lifting, $\oplus$-s-lifting, and SA-co-supplement modules, are introduced to emphasize the interplay between extending and lifting properties. Hybrid classes, including semi-rationally extending modules, are also considered, with an analysis of their closure properties and behavior over artinian, semiperfect, and group rings. Several open problems and conjectures are formulated, highlighting directions for future research in module theory and its applications.
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Copyright (c) 2026 Abdelkader Hamdouni
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