The Study Of Homological Property Of Rings And Modules Determined By Multiplicatively Closed Sets

In this thesis, we mainly discuss the homological property of rings and modules determined by multiplicatively closed sets. We study the properties of S-divisible modules, S-regular injective modules, S-Noether rings and S-Dedekind rings. Let R be a ring and M an R-module. Let S denote a regular multiplicative closed set of center in R. A left R-module M is said to be an S-divisible module if Ext1R(R/Ru, M) = 0 for any regular u ? S. A left ideal I is said to be S-regular if I? S ? (?). A left R-module E is said to be an S-regular injective if Ext1R(R/I,E) = 0 for any S-regular left ideal I of R. A commutative ring R is said to be S-Dedekind ring if every S-regular left ideal of R is invertible. It is proved that a conmutative ring R is S-Dedekind ring if and only if S-divisible modules are S-regular injective modules. Moreover, we prove that R is an S-Noether ring if and only if every sum of S-regular injective modules is S-regular injective. Let R be a conmmutative S-Noetherian ring and I be an S-regular ideal in R. We show that there are only a finite number of prime ideals minimal over I. A ring is said to be S-Noetherian if every S-regular ideal is finitely generated. At the same time,we also introduce the concepts of S-regular flat modules and S-coherent rings, and we prove that R is a left S-coherent ring if and only if any direct product of S-regular flat modules is S-regular flat.