ω-flat Modules And ω-weak Global Dimension Of Rings

In this paper, the methods of cohomology and w-operations are employedfor investigating w-?at modules and the w-?at dimension of modules. In thefirst chapter, the w-?at modules are characterized by using functor Tor and w-operations, and the ?at modules are proved to be w-?at modules. We take anexample to illustrate that there exists a w-?at module which is not a ?at module.Furthermore, we obtain some properties of w-?at modules, namely if R is aw-coherent rings, M is a w-?at module and finite presented type, then M is a w-projective module and if T is a w-linked over R, M is a R-module and w-?at, thenM is a T-module and W-?at. In the second chapter, the w-?at dimensionof modules and the w-global dimension of rings are considered with the methodsof cohomology. An example is taken to demonstrate that w-global dimension of aring is di?erent from global dimension of a ring. Moreover, the V N-regular ringsand PVMDs are characterized by the opinion of w-global dimension of rings. Itis shown that R is a V N-regular ring if and only if w.w.gl.dim(R) = 0; if andonly if every R-module is a w-?at module. If R is a domain, then R is a PVMDif and only if w.w.gl.dim(R) 1; which is equivalent to that every ideal of R isa w-?at ideal.