A Study Of Totally Reflexive Modules And The G-regularity Of Rings

Let R be a commutative Noetherian ring,g(R)denote the subcategory of R-mod consisting of all totally reflexive modules,and ε(R)denote the subcatego-ry of R-mod consisting of all R-modules M satisfying the inequality depth(Mm)≥depth(Rm)for any maximal ideal m.A Noetherian ring R is called G-regular if the local ring Rm is a G-regular local ring for any maximal ideal m of R,that is,any totally reflexive Rm-module is free;A ring homomorphism φ:R→S is called G-vanishing if Ext1R(g(R),S)=0 and Tor1R(gR),S)=0.In the thesis,we give some non-trivial examples of G-regular rings and G-vanishing homomorphisms respectively,investigate their some properties,and utilize these properties and combine some special orthogonal class to characterize some rings such as regular rings and Gorenstein rings.It is shown that totally reflexive modules satisfy ascent condition with respect to G-vanishing homomorphisms;a ring R is G-regular if the polynomial ring R[X]or R{X)is G-regular;a local ring(R,m)is either Gorenstein or G-regular if the maximal ideal m is decomposable;any two modules which are projectively equivalent over a local ring are either have the same depth or all in ε(R);R is a QF-ring if and only if any G-vanishing homomorphism over R is flat.Next,the G-regularity of commutative group rings and trivial extensions are dis-cussed respectively,which are used to consider some questions or construct some ex-amples.It is shown that any non-trivial local group ring R[G]is not G-regular,but non-local group ring R[G]maybe G-regular;R is an Iwanaga-Gorenstein ring if and only if any group ring R[G]is an Iwanaga-Gorenstein ring;the regularity and G-regularity of group rings over a field are consistent;any trivial extension R ∝ Ris not G-regular,and R ∝ M is G-regular if R is G-regular and the projection R ∝ M→R is G-vanishing.Finally,the G-regularity of local domains with Krull dimension one is considered.The structure with respect to(G-)regularity of one dimensional local domains with multiplicity three is given,and these rings is not G-regular if the number of minimal generators of the maximal ideal is equal to two.In addition,it is shown that an one dimensional local domain R is G-regular if and only if the embedding map R→Rc is G-vanishing,where Rc denotes integral closure of R.