Commutative rings in which every S-finite ideal is projective

In this paper, we introduce and explore a category of rings termed SFP-rings, defined by the property that each S-finite ideal of them is projective. We demonstrate that all hereditary rings inherently qualify as SFP-rings, while conversely, every SFP-ring is inherently semi-hereditary. Our research delves into the application and relevance of this concept across various facets of commutative ring extensions. These include localizations, direct products, trivial ring extensions, pullbacks, and the specific case of amalgamated duplication of a ring along an ideal. A significant outcome of our study is the identification and characterization of unique families of rings: those that are non-hereditary yet qualify as SFP-rings, and those that are semi-hereditary but do not meet the criteria for being SFP-rings. This work not only extends the theoretical understanding of ring structures but also contributes to the broader field of algebraic theory with practical examples and applications.