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On the Right Orthogonal Complement of the Class of ω-Flat Modules


Affiliations
1 Department of Mathematics, Faculty of Science, King Khalid University, P.O.Box 9004, Abha, Saudi Arabia
2 Department of Mathematics, Faculty of Science, University Moulay Ismail Meknes, Box 11201, Zitoune, Morocco
     

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Let R be a commutative ring. An R-module M is said to be ω-flat if TorR1 (M, N) is a GV-torsion R-module for all R-modules N. In this paper, we study the flat and projective dimensions of ω-flat modules. To do so, we study the elements of the right orthogonal complement of the class of all ω-flat modules, called ω-cotorsion modules, and we introduce and characterize the ω-cotorsion dimension for modules and rings. The relations between the ω-cotorsion dimension and other dimensions are discussed, and many illustrative examples are given. As applications, we give a new homological characterizations of PvMDs and a new upper bound on the global dimension of rings.
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  • On the Right Orthogonal Complement of the Class of ω-Flat Modules

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Authors

Fuad Ali Ahmed Almahdi
Department of Mathematics, Faculty of Science, King Khalid University, P.O.Box 9004, Abha, Saudi Arabia
Mohammed Tamekkante
Department of Mathematics, Faculty of Science, University Moulay Ismail Meknes, Box 11201, Zitoune, Morocco
Refat Abdelmawla Khaled Assaad
Department of Mathematics, Faculty of Science, University Moulay Ismail Meknes, Box 11201, Zitoune, Morocco

Abstract


Let R be a commutative ring. An R-module M is said to be ω-flat if TorR1 (M, N) is a GV-torsion R-module for all R-modules N. In this paper, we study the flat and projective dimensions of ω-flat modules. To do so, we study the elements of the right orthogonal complement of the class of all ω-flat modules, called ω-cotorsion modules, and we introduce and characterize the ω-cotorsion dimension for modules and rings. The relations between the ω-cotorsion dimension and other dimensions are discussed, and many illustrative examples are given. As applications, we give a new homological characterizations of PvMDs and a new upper bound on the global dimension of rings.

References