TY - JOUR AU - Suprapto, S AU - Wahyuni, Sri AU - Wijayanti, Indah Emilia AU - Irawati, I PY - 2011/07/31 TI - On τ [M ]-Cohereditary Modules JF - Jurnal ILMU DASAR; Vol 12 No 2 (2011) KW - Category τ [M ], Q-projective, P-injective, τ [M ]-cohereditary N2 - 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 ]. UR - https://jurnal.unej.ac.id/index.php/JID/article/view/66