On Φ-projective modules and Φ-Prüfer rings

Abstract

In this article, if S denotes a multiplicatively closed set of a commutative ring R, then S-fractional ideals and S-invertible ideals are introduced. Suppose that the nil radical Nil(R) of R is a divided prime ideal. Then ϕ-torsion free modules and ϕ-projective modules are introduced and it is shown that a finitely generated nonnil ideal I is ϕ-invertible if and only if I is ϕ-projective as R-module. So R is a ϕ-Prüfer ring if and only if each finitely generated nonnil ideal of R is ϕ-projective.