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 S w = { f ∈ R [ X ] | c ( f ) w = R } , where c ( f ) denotes the content of f. Let R { X } = R [ X ] S w and M { X } = M [ X ] S w 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 } is finitely generated projective over R { X } ; M is w-invertible if and only if M { X } is invertible over 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 Q 0 -PVMR, every finite type semi-regular module is w-projective.