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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, Yasser^2,3 (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]
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Database:	Engineering Source

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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]
ISSN:	02194988
DOI:	10.1142/S0219498826503081