On Modules For Which Every Cosingular Verify The D4-Condition

Let R be an associative ring with unity and M an unital left R-module. In this paper we introduce D41-module which is a generalization of D4-module. A module M is called D41 if M = N ⊕ K with N,K ≤ M, K is cosingular and f∶ N → K is an epimorphism, then is a direct summand of N. Some basic properties of these modules are investigated. It is shown that the class of rings R over which a D41-module is a D4-module is exactly that of COSP-rings. Also, we study the relations between D41-module and other related modules.