TAILIEUCHUNG - Báo cáo toán học: "When M-Cosingular Modules Are Projective"

M là một R-mô-đun. Talebi và Vanaja điều tra loại σ [M] như vậy mà mỗi mô-đun M-cosingular trong σ [M] là projective trong σ [M]. Trong ánh sáng của tài sản này, chúng tôi gọi M một COSP-module nếu mỗi mô-đun M-cosingular projective trong σ [M]. | Vietnam Journal of Mathematics 33 2 2005 214-221 V Í e It ini ai m J o mt r im ai I of MATHEMATICS VAST 2005 When M-Cosingular Modules Are Projective Derya Keskin Tutuncu1 and Rachid Tribak Department of Mathematics University of Hacettepe 06532 Beytepe Ankara Turkey Departement de Mathématiques Université Abdelmalek Essaâdi Faculte des Sciences de Tétouan . Tétouan Morocco Received September 11 2004 Revised April 4 2005 Abstract. Let be an -module. Talebi and Vanaja investigate the category such that every -cosingular module in is projective in . In the light of this property we call a COSP-module if every -cosingular module is projective in . This note is devoted to the investigation of these classes of modules. We prove that every COSP-module is a coatomic module having a semisimple radical. We also characterise COSP-module when every injective module in is amply supplemented. Finally we obtain that a COSP-module is artinian if and only if every submodule has finite hollow dimension. 1. Introduction . . . coclosed small . .supplement supplemented. . When -Cosingular Modules Are Projective 215 amply supplemented. .M. .cosingular non- -cosingular . .COSP-. 2. Results . Example . - - kZ .-k--Z kZ. Z Z . Z kZ. . . . . . . .S. Example . . . . . . . . Proposition . Let be a COSP-module. Then the following statements are true. . Every -small module is semisimple. . For every module. Rad . Soc . Proof. . N . . . . N . . . K . . . . . . iE I . Proposition . Let be a module. Then is COSP if and only if every module in is COSP. 216 Derya Keskin Tutuncu and Rachid Tribak In particular any submodule homomorphic image and direct sum of COSP-modules are again COSP. Proof. . Example . . Proposition . Let be a COSP-module. Then every module. has a maximal submodule. Proof. .coatomic. . Theorem . Let be a COSP-module and. Then every nonzero submodule of is coatomic. Proof. . . Example kZ Corollary . Let Rad. be a COSP-module. Then

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