FP-injective , injective and pure R-module

Question

prove that : FP-injective R-modules, which are pure-injective, are injective ? I know that an R-module M is called FP-injective if Ext1(N,M) = 0 for all finitely presented R-modules N and R-submodule N of an R-module M is called pure in M if for all (finitely presented) R-modules F, the map : F⊗N → F⊗M induced by the inclusion map N → M is injective. and an exact sequence 0 → N → M → L → 0 is called a pure-exact sequence if Im(N→M ) is pure in M. and R-module M is called pure-injective if it has the injectivity property relative to all pure-exact sequences but i cant got the proof ?

Answer 1

FP injective module Is a module that the embedding in its injective envelope Is pure ; pure injective module Is a module can be lift by pure extension . then we consider the identity homomorphism of M and it can be lift by the pure embedding to its envelope and thus it Is a direct snmmand of injective module and thus a injective module