P-Dense Submodules

Let m and p be right r-modules. a submodule k ofan r-module m is called p-dense if for each m e m, (k : m) isa p-faithful right ideal of r. pr is nonsingular if and only if, foreach r-module m, every essential submodule of m is a p-densesubmodule. for any r-module m, we obtain p-rational extentionof m and equivalent condition in order that m is equal withits p-rational extention is found. an r-module p is called rightkasch if every simple r-module can be embedded in p. finally,we given some equivalent conditions for an r-module p to beright kasch.