P-Dense Submodules

Let M and P be right R?modules. A submodule K of an R?module M is called P?dense if for each m 2 M; (K : m) is a P?faithful right ideal of R: PR is nonsingular if and only if, for each R?module M; every essential submodule of M is a P?dense submodule. For any R?module M; we obtain P?rational exten- tion of M and equivalent condition in order that M is equal with its P?rational extention is found. An R?module P is called right Kasch if every simple R?module can be embedded in P: Finally, we given some equivalent conditions for an R?module P to be right Kasch.