On Absolutely FP-neat Modules And Absolutely FP-s-pure Modules

In this dissertation,we combine the purity and neatness of modules to introduce and investigate absolutely FP-neat modules and absolutely FP-s-pure modulesIn the second chapter,we define absolutely FP-neat modules,give some equiva-lent characterizations of absolutely FP-neat modules,prove that a left R-module M is absolutely FP-neat module if and only if ExtR1(S.M)=0 for every finitely presented simple left R-modules S,obtain that the concept of absolutely FP-neat modules is a proper generalization of absolutely neat modules and absolutely pure modules,that is,every maximal left ideal of a ring R is finitely generated(i.e.,R is a left N-ring)if and only if every absolutely FP-neat left R-module is absolutely neat.Moreover,we introduce FP-max-flat modules,whose character modules are absolutely FP-neat,ob-tain the closeness of the class of FP-max-flat modules under extensions,direct sums,direct summands and pure submodules.We also define left FP-max-coherent rings,left FP-max-hereditary rings and left FP-max-regular rings,which are characteized by FP-neat modules and FP-max-flat modulesIn the third chapter,we restrict simple modules in the concept of absolutely s-pure modules to finitely represented simple modules and introduce absolutely FP-s-pure modules.We give some equivalent characterizations of absolutely FP-s-pure modules,prove that a right R-module M is absolutely FP-s-pure module if and only if ExtR1(Tr(S),M)=0 for every finitely represented simple left R-module S,and then obtain the closeness of the class of absolutely FP-s-pure modules under extensions,di-rect sums and direct summands.Moreover,we introduce FP-neat-flat modules,whose character modules are absolutely FP-s-pure modules,give some equivalent character-izations of FP-neat-flat modules.Finally,we use absolutely FP-s-pure modules and FP-neat-flat modules to characterize left Σ-CS rings and left FP-Kasch ringsBase on previous chapters,in the fourth chapter,we obtain the existence of FP-max-flat-preenvelopes and absolutely FP-s-pure covers,that is,every right R-module has an FP-max-flat-preenvelope if and only if R is a left FP-max-coherent ring,if and only if every left R-module has an FP-neat-cover,and prove that every right R-module has a monic FP-max-flat-preenvelope if and only if R is a left FP-max-FC ring,if and only if every left R-module has an epic FP-neat cover,while every right R-module has an epic FP-max-flat-preenvelope if and only if R is a left FP-max-hereditary ring,if and only if every left R-module has a monic FP-neat cover.Moreover,we show that absolutely FP-s-pure-preenvelopes and FP-s-pure-covers alway exist for every right R-module,and obtian the sufficient and necessary conditions for the existence of monic(epic)FP-s-pure-covers.We also prove absolutely FP-max-flat-preenvelopes and FP-max-flat-covers alway exist for every left R-modules,and that every left R-module has an epic FP-neat-envelope if and only if every finitely represented simple left ideal of R is projective.