The Study Of N-torsionfree And N-divisible Modules Over Noncommucative Rings

In this thesis,some basic properties of n-torsionfree and n-divisible modules are studied.And left n-P-coherent rings are defined as the rings such that every direct product of n-torsionfree right R-modules yields a n-torsionfree module.We obtained that a ring R is left n-P-coherent,if and only if every right R-module has n-torsionfree preenvelope,if and only if every left R-module has n,-divisible cover.It is proved that a ring R is left coherent,if and only if it is left n-P-coherent for any integer n ? 1.Some results about coherent rings become corollaries of this dissertation by using the relation between n-torsionfree and flat right R-moduies,and the relation between left n-P-coherent and left coherent rings.Moreover,Dn-injective modules,Dn-flat modules and TFn-projective modules are found when we study n-torsionfree preenvelope and n-divisible precover.