Strongly N-FP-injective Modules And(N,n)-projective Modules

In this dissertation,we introduce and investigate strongly n-FP-injective mod-ules and dimensions,use strongly n-FP-injective dimensions to define(n,n)-projective modules and dimensions,study some properties and characterizations of these mod-ules and dimensions,invariant properties under an almost excellent extension of rings,Morita duality and existences of(pre)envelopes and(pre)covers.In the first chapter,we mainly investigate strongly n-FP-injective modules and strongly n-FP-injective dimension of rings and modules.We first introduce strongly n-FP-injective modules,give the closeness of the class of strongly n-FP-injective mod-ules under extensions,direct products,direct summands and cokernels,obtain that the class of strongly n-FP-injective modules is coresolving,prove that the class of strong-ly n-FP-injective modules is a proper generalization of the class of n-FP-injective modules and give equivalent characterizations of right n-coherent rings by(strong-ly)n-FP-injective right R-modules.Moreover,we introduce n-presented right global dimensions of rings,define right n-regular rings and right n-heretidary rings,and characterize these rings and the dimensions by(strongly)n-FP-injective modules and n-flat modules.Under an almost excellent extension of rings,we study properties of strongly n-FP-injective modules,n-flat modules,n-presented right global dimensions of rings and right n-regular rings.Denote the class of strongly n-FP-injective mod-ules by S F-PnI,we prove that(?SFPnI,SFPnI)is a complete cotorsion pair and so every right R-module has a special S FFPnI-preenvelope.If R is a right n-coherent ring,then every right R-module has a SFPnI-cover.Moreover we obtain some e-quivalent characterizations between existences of surjective(injective)SFPnI-covers and right n-regular rings,right n-heretidary rings.Dual to the concept of strongly n-FP-injective modules,we introduce strongly n-CP-projective modules,prove that the class of strongly n-CP-projective modules is resolving,give some equivalent char-acterizations of right CP-hereditary rings,and proved that strongly n-FP-injective right R-modules and strongly n-CP-projective right R-modules is a Morita dual pair.Moreover we define strongly n-FP-injective dimensions of right R-modules and right strongly n-FP-injective global dimensions of rings,give several equivalent characteri-zations of strongly n-FP-injective dimension of modules,get some relations between right strongly n-FP-injective global dimensions and n-presented right global dimen-sion of rings.Let TIn be the class of right R-modules with s-n-fp-id(N)?n,we prove(?FIn,FIn)is a complete cotorsion pair and that every right R-module has a special FIn-envelope.Moreover we prove that r.s-n-FP-ID(R)? if and only if every right R-module has a FIn-preenvelope which has unique mapping Property if and only if every right R-module has an injective FIn-cover.Base on previous chapter,in the second chapter,we use strongly n-FP-injective dimensions to define(n,n)-projective right R-modules,obtain the closeness of the class of(n,n)-projective right R-modules under extensions,direct sums,direct summands.Further,we define(n,n)-projective dimensions of modules and(n,n)-projective right global dimension of rings,obtain some equivalent characterizations of the given dimen-sions,define and characterize right(n,n)-hereditary rings and(n,n)-semisimple rings,give equivalent conditions that every(n,n)-flat right R-module is(n,n)-projective for a left n-coherence and right(n,n)-hereditary ring.Under an almost excellent exten-sion of rings,we prove some invariant properties of strongly n-FP-injective dimensions,(n,n)-projective modules,(n,n)-projective dimensions.