Fractions And Gorenstein Modules

The fractions are very essential in commutative algebra,which promote the development and prosperity of commutative algebra.With the development of the homological algebra and commutative algebra,the intersections between them become more.The definition of Gorenstein rings was given by Auslander and Bridger.Later,Enochs and Jenda defined Gorenstein projective modules over an arbitrary ring.Since then Gorenstein homological algebra became fruitful.But most scholars focus on the homological properties and characteristics of Gorenstein modules.But there are very few research on the combination of Gorenstein modules and fractions of rings in commutative algebra.Therefore,the combination between fractions and Gorenstein modules are worth studying.In this thesis,we firstly study when the fractions of the residue class rings can be domains.We prove the following resultTheorem 1 The fraction of zn can be a field if and only if there is a prime number p such that p is a divisor of n and p2 is not a divisor of nOn the other hand,Gorenstein modules are the important objects in Gorenstein homology.In this thesis,we study the properties of Gorenstein modules in terms of fractions of rings in commutative algebra,and show whether the fractions of rings keep Gorenstein modules.We get the following results:Theorem 2 Let M be a Gorenstein flat R-module and S any multiplicatively subset of R.Then S-1M is Gorenstein flat if and only if ToriS-1R(F,S-1M)=0 for any injective S-1R-module F and i≥1.Theorem 3 Let R be an n-Gorenstein ring,S any multiplicatively subset of R and G be a R-module.If G is a Gorenstein injective R-module,then S-1G is a Gorenstein injective S-1R-moduleTheorem 4 Let R be a Noetherian ring,S any multiplicatively subset of R and M be finite generated R-module.If M is a Gorenstein projective R-module,then S-1M is a Gorenstein projective S-1R-module.