Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules
In this paper, we investigate the notions of X⊥-projective, X-injective, and X-flat modules and give some characterizations of these modules, where X is a class of left modules. We prove that the class of all X⊥-projective modules is Kaplansky. Further, if the class of all X-injective R-modules is c...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/8072134 |
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| author | Arunachalam Umamaheswaran Ramalingam Udhayakumar Chelliah Selvaraj Kandhasamy Tamilvanan Masho Jima Kabeto |
| author_facet | Arunachalam Umamaheswaran Ramalingam Udhayakumar Chelliah Selvaraj Kandhasamy Tamilvanan Masho Jima Kabeto |
| author_sort | Arunachalam Umamaheswaran |
| collection | DOAJ |
| description | In this paper, we investigate the notions of X⊥-projective, X-injective, and X-flat modules and give some characterizations of these modules, where X is a class of left modules. We prove that the class of all X⊥-projective modules is Kaplansky. Further, if the class of all X-injective R-modules is contained in the class of all pure projective modules, we show the existence of X⊥-projective covers and X-injective envelopes over a X⊥-hereditary ring. Further, we show that a ring R is Noetherian if and only if W-injective R-modules coincide with the injective R-modules. Finally, we prove that if W⊆S, every module has a W-injective precover over a coherent ring, where W is the class of all pure projective R-modules and S is the class of all fp−Ω1-modules. |
| format | Article |
| id | doaj-art-310ccf8e8d6a4a439076c7b082e4e468 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-310ccf8e8d6a4a439076c7b082e4e4682025-02-03T06:11:53ZengWileyJournal of Mathematics2314-47852022-01-01202210.1155/2022/8072134Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of ModulesArunachalam Umamaheswaran0Ramalingam Udhayakumar1Chelliah Selvaraj2Kandhasamy Tamilvanan3Masho Jima Kabeto4Department of MathematicsDepartment of MathematicsDepartment of MathematicsDepartment of MathematicsDepartment of MathematicsIn this paper, we investigate the notions of X⊥-projective, X-injective, and X-flat modules and give some characterizations of these modules, where X is a class of left modules. We prove that the class of all X⊥-projective modules is Kaplansky. Further, if the class of all X-injective R-modules is contained in the class of all pure projective modules, we show the existence of X⊥-projective covers and X-injective envelopes over a X⊥-hereditary ring. Further, we show that a ring R is Noetherian if and only if W-injective R-modules coincide with the injective R-modules. Finally, we prove that if W⊆S, every module has a W-injective precover over a coherent ring, where W is the class of all pure projective R-modules and S is the class of all fp−Ω1-modules.http://dx.doi.org/10.1155/2022/8072134 |
| spellingShingle | Arunachalam Umamaheswaran Ramalingam Udhayakumar Chelliah Selvaraj Kandhasamy Tamilvanan Masho Jima Kabeto Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules Journal of Mathematics |
| title | Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules |
| title_full | Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules |
| title_fullStr | Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules |
| title_full_unstemmed | Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules |
| title_short | Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules |
| title_sort | existence of covers and envelopes of a left orthogonal class and its right orthogonal class of modules |
| url | http://dx.doi.org/10.1155/2022/8072134 |
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