Two generalizations of projective modules and their applications

Let R be a commutative ring, M be an R-module, and w be the so-called w-operation on R  . Set Sw={f∈R[X]|c(f)w=R}Sw={f∈R[X]|c(f)w=R}, where c(f)c(f) denotes the content of f  . Let R{X}=R[X]SwR{X}=R[X]Sw and M{X}=M[X]SwM{X}=M[X]Sw be the w-Nagata ring of R and the w-Nagata module of M respectively. Then we introduce and study two concepts of w-projective modules and w-invertible modules, which both generalize projective modules. To do so, we use two main methods of which one is to localize at maximal w-ideals of R and the other is to utilize w-Nagata modules over w-Nagata rings. In particular, it is shown that an R-module M is w  -projective of finite type if and only if M{X}M{X} is finitely generated projective over R{X}R{X}; M is w  -invertible if and only if M{X}M{X} is invertible over R{X}R{X}. As applications, it is shown that R is semisimple if and only if every R-module is w  -projective and that, in a Q0Q0-PVMR, every finite type semi-regular module is w-projective.