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.