On w-faithfully flat algebras

In this paper, we characterize the w-faithfully flat algebra by the w-module theory, and we prove that a w-flat w-linked R-algebra (T,φ) is w-faithfully flat if and only if θ:wφ-Spec(T)→w-Spec(R) is surjective, where wφ-Spec(T) is the set of prime wφ-ideals of T, and w-Spec(R) is the set of prime w-ideals of R. Let K be a field and let R=K[x,y,u,v] with xv=yu defined over the field K. Set T=K[x,y,u/x] and q=xT. It is well-known that T as R-module is not flat and hence the polynomial ring T[Z] as R-modules is flat. But we prove that T[Z]q[Z] is a faithfully flat Rp-algebra where p=q⋂R.