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 G(w) = {f is an element of R[X] vertical bar c(f)(w) = R}, where c(f) denotes the content of f. Let R{X} = R[X]G(w) and M{X} = M[X]G(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. (C) 2014 Elsevier B.V. All rights reserved.