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Dual perspectively decomposable modules.

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Title: Dual perspectively decomposable modules.
Authors: Das, Soumitra1 (AUTHOR) soumitrad330@gmail.com, Ibrahim, Yasser2,3 (AUTHOR) yabdelwahab@taibahu.edu.sa, Taşdemi̇r, Özgür4 (AUTHOR) ozgurtasdemir@trakya.edu.tr, Yousif, Mohamed5 (AUTHOR) yousif.1@osu.edu
Source: Journal of Algebra & Its Applications. Aug2026, Vol. 25 Issue 9, p1-19. 19p.
Subjects: Modules (Algebra), Indecomposable modules
Abstract: A module M is called dual perspectively indecomposable if, M does not contain proper perspectively related submodules A and B with A + B = M , where two submodules A and B of M are called perspectively related, and denoted by A ∼ B , if M = A ⊕ C = B ⊕ C , for a submodule C ⊆ M. Every indecomposable module is dual perspectively indecomposable, but the converse is not true. Moreover, M is called dual perspectively decomposable (dual PD-module) if, A ∩ B = 0 for every pair of proper submodules A and B of M with A ∼ B and A + B = M. Examples are provided to show that the class of dual P D -modules lies strictly between the classes of summand-dual-square-free and D 4 -modules. We will show that every dual P D -module is a finite direct sum of dual perspectively indecomposable submodules. As an application, we prove that if M is a dual P D -module with the finite exchange, then M is clean and has the full exchange. This is a partial answer to Crawley–Jónsson's open question that asks whether the finite exchange property of a module implies the full exchange property. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Algebra & Its Applications is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: A module M is called dual perspectively indecomposable if, M does not contain proper perspectively related submodules A and B with A + B = M , where two submodules A and B of M are called perspectively related, and denoted by A ∼ B , if M = A ⊕ C = B ⊕ C , for a submodule C ⊆ M. Every indecomposable module is dual perspectively indecomposable, but the converse is not true. Moreover, M is called dual perspectively decomposable (dual PD-module) if, A ∩ B = 0 for every pair of proper submodules A and B of M with A ∼ B and A + B = M. Examples are provided to show that the class of dual P D -modules lies strictly between the classes of summand-dual-square-free and D 4 -modules. We will show that every dual P D -module is a finite direct sum of dual perspectively indecomposable submodules. As an application, we prove that if M is a dual P D -module with the finite exchange, then M is clean and has the full exchange. This is a partial answer to Crawley–Jónsson's open question that asks whether the finite exchange property of a module implies the full exchange property. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Algebra & Its Applications is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1142/S0219498826503081
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      – Code: eng
        Text: English
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        PageCount: 19
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      – SubjectFull: Indecomposable modules
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              M: 08
              Text: Aug2026
              Type: published
              Y: 2026
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