The Study Of The W-weak Global Dimension Of Pullbacks

Let R(?)T be a w-linked extension of domains.In Chapter 1,we introduce the concept of wR-flat dimension of a T-module M(wR-fdTM for short).Then we introduce the concept of wR-weak global dimension of ring T(wR-w.gl.dim(T)for short).The wR-weak global dimension of ring T can be used to characterize fields and PwRMDs.More precisely,if T is w-linked over R,then T is a field if and only if wR-w.gl.dim(T)=0,and T is a PwRMD if and only if wR-w.gl.dim(T)?1.Let R be a commutative ring with identity,and let I0 be a nonzero ideal of R.In Chapter 2,we introduce the c-operation over the factor ring R=R/I0,then we introduce the c-flat module over R,the c-flat dimension of an R-module M,and the c-weak global dimension of R.As an application,for a Cartesian square(RDTF,M),we give an upper bound for the w-weak global dimension of R by the wR-weak global dimension of T and the c-weak global dimension of D.More precisely,let(RDTF,M)be a Cartesian square,where T is w-linked over R,then we have w-w.gl.dim(R)?max{wR-w.gl.dim(T)+w-fdR.T,c-w.gl.dim(D)+W-fdRD}.Furthermore,for a Milnor square(RDTF,M),we can get that w-w.gl.dim(R)?max{wR-w.gl.dim T)+w-fdRT,w-w.gl.dim(D)+w-fdR.D}.Then,in Chapter 4,we give some results of pullbacks of locally pseudo-valuation domains(LPVDs for short)and t-locally pseudo-valuation domains(t-LPVDs for short).Let(RDTF,M)be a Milnor square of type ?.We prove that R is an LPVD if and only if D and T are LPVDs and TM is a VD.Let(RDTF,M)be a Milnor square of type ?.We prove that R is an LPVD if and only if D is a field and T is an LPVD.Let(RDTF,M)be a Milnor square of type ?.We prove that R is a t-LPVD if and only if D and T are t-LPVDs and TM is a VD.Let(RDTF,M)be a Milnor square of type ?.We prove that R is a t-LPVD if and only if D is a field,T is a t-LPVD and TM is a PVD.In addition,we also study the homological properties of LPVDs and t-LPVDs.More precisely,let R be an LPVD(resp.,a t-LPVD)but not a Prufer domain(resp.,PvMD).Then w.gl.dim(R)(resp.,w-w.gl.dim(R))=2 or ?:(1)w.gl.dim(R)(resp.,w-w.gl.dim(R))=2 if and only if MRM=M2RM for any M ? Max(R)(resp.,w-Max(R))with RM not a VD.(2)w.gl.dim(R)(resp.,w-w.gl.dim(R))=? if and only if MRM?M2RM for some M ? Max(R)(resp.,w-Max(R))with RM not a VD.Finally,in Chapter 5,we introduce the concept of w-Milnor squares and study some properties of w-Milnor squares.Thus,we give some rings with w-weak global dimensions equal to 2 by using w-Milnor square.Namely,let(RDTF,M)be a w-Milnor square,where D is a field,and T is a PWRMD.Then w-w.gl.dim(R)=2 or ? depending on M=M2 or not,respectively.