On τ [M ]-Cohereditary Modules
Let R be a ring with unity and N a left R-module. Then N is said linearly independent to R (or N is R-linearly independent) if there exists a monomorphism φ : R(Λ) → N . We can define a generalization of linearly independency relative to an R-module M. N is called M-linearly independent if there exi...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Fakultas MIPA Universitas Jember
2011-07-01
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| Series: | Jurnal Ilmu Dasar |
| Subjects: | |
| Online Access: | https://jurnal.unej.ac.id/index.php/JID/article/view/66 |
| _version_ | 1818693125753798656 |
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| author | S Suprapto Sri Wahyuni Indah Emilia Wijayanti I Irawati |
| author_facet | S Suprapto Sri Wahyuni Indah Emilia Wijayanti I Irawati |
| author_sort | S Suprapto |
| collection | DOAJ |
| description | Let R be a ring with unity and N a left R-module. Then N is said linearly independent to R (or N is R-linearly independent) if there exists a monomorphism φ : R(Λ) → N . We can define a generalization of linearly independency relative to an R-module M. N is called M-linearly independent if there exists a monomorphism φ:M(Λ) →N. Amodule iscalled M-sublinearly independentif is a factormodule of a module which is M-linearly independent. The set of M-sublinearly independent modules is denoted by τ [M ]. It is easy to see that τ [M ] is subcategory of category R-Mod. Furthermore, any submodule, factor module and external direct sum of module in τ [M ] are also in τ [M ]. A module is called τ [M ]-injective if it is P-injective, for all modules P in τ [M ]. Q is called τ [M ]-cohereditary if Q ∈τ [M ] and any factor module of Q is τ [M ]-injective. In this paper, we study the characterization of category τ [M ]-cohereditary modules. For any Q in τ [M ], Q is a τ [M ]-cohereditary if and only if every submodule of Q-projective module in τ [M ] is Q-projective. Moreover, Q is a τ [M ]-cohereditary if and only if every factor module of Q is a direct summand of module which contains this factor module. Also, we obtain some cohereditary properties of category τ [M ]. There are: for any R-modules P, Q. If Q is P-injective and every submodule of P is Q-projective, then Q is cohereditary (1); if P is Q-projective and Q is cohereditary, then every submodule of P is Q-projective (2); a direct product of modules which τ [M ]-cohereditary is τ [M ]-cohereditary (3). The cohereditary characterization and properties of category τ [M ] above is truly dual of characterization and properties of category τ [M ]. |
| first_indexed | 2024-12-17T13:08:43Z |
| format | Article |
| id | doaj.art-299965247fba43368915f9fbc71270de |
| institution | Directory Open Access Journal |
| issn | 1411-5735 2442-5613 |
| language | English |
| last_indexed | 2024-12-17T13:08:43Z |
| publishDate | 2011-07-01 |
| publisher | Fakultas MIPA Universitas Jember |
| record_format | Article |
| series | Jurnal Ilmu Dasar |
| spelling | doaj.art-299965247fba43368915f9fbc71270de2022-12-21T21:47:11ZengFakultas MIPA Universitas JemberJurnal Ilmu Dasar1411-57352442-56132011-07-0112218419066On τ [M ]-Cohereditary ModulesS SupraptoSri WahyuniIndah Emilia WijayantiI IrawatiLet R be a ring with unity and N a left R-module. Then N is said linearly independent to R (or N is R-linearly independent) if there exists a monomorphism φ : R(Λ) → N . We can define a generalization of linearly independency relative to an R-module M. N is called M-linearly independent if there exists a monomorphism φ:M(Λ) →N. Amodule iscalled M-sublinearly independentif is a factormodule of a module which is M-linearly independent. The set of M-sublinearly independent modules is denoted by τ [M ]. It is easy to see that τ [M ] is subcategory of category R-Mod. Furthermore, any submodule, factor module and external direct sum of module in τ [M ] are also in τ [M ]. A module is called τ [M ]-injective if it is P-injective, for all modules P in τ [M ]. Q is called τ [M ]-cohereditary if Q ∈τ [M ] and any factor module of Q is τ [M ]-injective. In this paper, we study the characterization of category τ [M ]-cohereditary modules. For any Q in τ [M ], Q is a τ [M ]-cohereditary if and only if every submodule of Q-projective module in τ [M ] is Q-projective. Moreover, Q is a τ [M ]-cohereditary if and only if every factor module of Q is a direct summand of module which contains this factor module. Also, we obtain some cohereditary properties of category τ [M ]. There are: for any R-modules P, Q. If Q is P-injective and every submodule of P is Q-projective, then Q is cohereditary (1); if P is Q-projective and Q is cohereditary, then every submodule of P is Q-projective (2); a direct product of modules which τ [M ]-cohereditary is τ [M ]-cohereditary (3). The cohereditary characterization and properties of category τ [M ] above is truly dual of characterization and properties of category τ [M ].https://jurnal.unej.ac.id/index.php/JID/article/view/66category τ [m ]q-projective, p-injectiveτ [m ]-cohereditary |
| spellingShingle | S Suprapto Sri Wahyuni Indah Emilia Wijayanti I Irawati On τ [M ]-Cohereditary Modules Jurnal Ilmu Dasar category τ [m ] q-projective, p-injective τ [m ]-cohereditary |
| title | On τ [M ]-Cohereditary Modules |
| title_full | On τ [M ]-Cohereditary Modules |
| title_fullStr | On τ [M ]-Cohereditary Modules |
| title_full_unstemmed | On τ [M ]-Cohereditary Modules |
| title_short | On τ [M ]-Cohereditary Modules |
| title_sort | on τ m cohereditary modules |
| topic | category τ [m ] q-projective, p-injective τ [m ]-cohereditary |
| url | https://jurnal.unej.ac.id/index.php/JID/article/view/66 |
| work_keys_str_mv | AT ssuprapto ontmcohereditarymodules AT sriwahyuni ontmcohereditarymodules AT indahemiliawijayanti ontmcohereditarymodules AT iirawati ontmcohereditarymodules |