| In this dissertation,we study w-homological theory determined by the class of Pw∞ modules and its applications in multiplicative ideal theory.In Chapter 1,we introduce and study w∞-projective modules.We prove that(Pw∞,Pw∞⊥)is a complete hereditary coto-sion theory,where Pw∞ denotes the class of all w∞-projective R-modules.So we introduce and study w∞-projective dimension of modules and of rings.Using these concepts and prop-erties,we give a new w-homological characterisation of semisimple rings and Krull domains.And,using these concepts and properties,we show that,for a Krull domain R with pdRQ=2,Pw(?)∞don’t contain W∞.So we prove that the classes of Pw(?)∞and W∞ in[58]are different.In Chapter 2,we introduce and study w-Matlis cotorsion modules.Meanwhile,we study the rela-tionship among w-Matlis cotorsion strong w-modules and w-cotorsion w-modules and Warfield cotorsion strong w-modules.So we prove that if D is a torsion hw∞-divisible R-module with w∞-pdRD≤1,then D is direct summand of K(?)R F for some w∞-projective w-module F over R.In Chapter 3,to give a negative answer to Lee’s open question in[27],we investigate properties of 1-perfect domains in Milnor squares.For a Milnor square(RDTF,M),in this chapter,it is proved that R is a 1-perfect domain if and only if both D and T are 1-perfect domains;R is an almost perfect domain if and only if D is a field and T is an almost perfect domain;R is a Matlis domain if and only if T is a Matlis domain.Furthermore,to give a negative answer to Lee’s question in[27],we construct a counter example which is a 1-perfect domain R with w.gl.dim(R)=∞. |
