P-Dense Submodules

Abstract

Let M and P be right R−modules. A submodule K of an R−module M is called P−dense if for each m ∈ 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 extention 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.