A note on an R –module with (m, n)-pure intersection property

Let R be a ring. Given two positive integers m and n , an R moduleV is said to be (m,n)-presented, if there is an exact sequence of R -modules 0→K →R^m →V →0 m with K is n -generated. A submodule N of a right R - module M is said to be (m,n)-pure in M , if for every (m,n)-Presented left R - module V the canonical map N bigotimesV →M bigotimes R V is a monomorphism. An R -module M has the (m,n)-pure intersection property if the intersection of any two (m,n)-pure submodules is again (m,n)-pure. In this paper we give some characterizations, theorems and properties of modules with the (m,n)-pure intersection property.