New generalizations of P-extending modules

In this study, E is unitary right module over associative ring S with unit element. A sub module K of m dule E is called in E and noted. by (K ≤ r E) , if for each, y∊ E and x∊ E /{0}, ∊ ∊ [K.R.Goodearl 1976]. In this work, we used the rational sub module to introduce and study a new generalization of the class of P-extending modules namely rationally principally extending module (for shortly, RP-extending) where an S- E is said to be RP- xtending ; if every cyclic sub m dule of E is rational in diret summand of E.On the other hand, we present a new class of modules namely Rc-sub modules and Rc-modules. Several properties and characterizations of these notions are given. Also we study the relation with our notions and some related well known concepts. In the last part of this study, a new generalization of injectivity are introduced and studied by using Rc-sub modules. The nations of (quasi-) injective modules is generalized to that of Rcc- (quasi-)injective modules: Let W and be -modules. Then is called W-injective if every homomorphism , where is Rc-closed submodule of W can be extended to an -homomorphism . An -module is called –quasi-injective or -self- injective, if is -injective..Numerous properties and characterizations of these generalizations are given. Moreover, the relation between these notions is studied.