Researchs On Homological Methods Determined By Regular Ideals

Let R be a commutative ring,T be a set of finitely generated regular ideals of R.In this thesis,we introduce the concepts of regular coflat modules and regular flat modules.An R-module E is called regular coflat if ExtR1(R/I,E)=0 for any I∈T.An R-module M is called regular flat if Tor1R(M,R/I)=0 for any regular ideal of R.Meanwhile,we discuss their properties and generalize Chase’s theorem on coherent rings by using regular flat modules.And then,we characterize Prufer rings by using the relationship of regular coflat modules and divisible modules.The main results are as follows:firstly,we give some equivalent characteriza-tions of regular coflat modules and regular flat modules.Secondly,we introduce the concept of regular coherent rings.A ring R is called regular coherent if I is finitely presented for any I∈T T.We prove that R is a regular coherent ring if and only if any direct product of(regular)flat modules is regular flat.In particular,we show that Prufer rings are regular coherent rings.And we prove that R is a Prufer ring if and only if all divisible modules are regular coflat modules if and only if factor modules of regular coflat modules are regular coflat modules.Fi-nally,we define the concepts of the global regular coflat dimension(reg-CFD(R))and the weak global regular flat dimension(reg-FD(R)).We prove that if RCF(the class of all regular coflat modules)satisfies the condition that the cokernel of injective homomorphism is closed,then reg-CFD(R)=reg-FD(R)and in this case,reg-CFD(R)≤1 if and only if R is a Priifer ring.